Forcing (mathematics)From Wikipedia, the free encyclopedia

Contents

1 Amoeba order 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Boolean-valued model 22.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Interpretation of other formulas and sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Boolean-valued models of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Relationship to forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.4.1 Boolean-valued models and syntactic forcing . . . . . . . . . . . . . . . . . . . . . . . . . 42.4.2 Boolean-valued models and generic objects over countable transitive models . . . . . . . . . 4

2.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Cantor algebra 63.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Cohen algebra 74.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5 Collapsing algebra 85.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

6 Complete Boolean algebra 96.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.2 Properties of complete Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.3 The completion of a Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.4 Free κ-complete Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

7 Continuum hypothesis 127.1 Cardinality of infinite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127.2 Independence from ZFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.3 Arguments for and against CH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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7.4 The generalized continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.4.1 Implications of GCH for cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . 15

7.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8 Countable chain condition 188.1 Examples and properties in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

9 Easton’s theorem 209.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.2 No extension to singular cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

10 Forcing (mathematics) 2210.1 Intuitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210.2 Forcing posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

10.2.1 P-names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.2.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

10.3 Countable transitive models and generic filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.4 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.5 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.6 Cohen forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.7 The countable chain condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.8 Easton forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.9 Random reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.10Boolean-valued models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.11Meta-mathematical explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.12Logical explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.15External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

11 Generic filter 3011.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

12 Iterated forcing 3112.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

13 Laver property 33

CONTENTS iii

13.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

14 List of forcing notions 3414.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.3 Amoeba forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.4 Cohen forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.5 Grigorieff forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.6 Hechler forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.7 Jockusch–Soare forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.8 Iterated forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.9 Laver forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.10Levy collapsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.11Magidor forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614.12Mathias forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614.13Namba forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614.14Prikry forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614.15Product forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3714.16Radin forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3714.17Random forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3714.18Sacks forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3714.19Shooting a fast club . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3714.20Shooting a club with countable conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3714.21Shooting a club with finite conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.22Silver forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.23References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.24External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

15 Martin’s maximum 3915.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

16 Nice name 4016.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4016.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

17 Proper forcing axiom 4117.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4117.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4117.3 Consistency strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4117.4 Other forcing axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4117.5 The Fundamental Theorem of Proper Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4217.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4217.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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18 Ramified forcing 4318.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

19 Random algebra 4419.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

20 Rasiowa–Sikorski lemma 4520.1 Proof of the Rasiowa–Sikorski lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4520.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4520.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4520.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4620.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

21 Sacks property 4721.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

22 Sunflower (mathematics) 4822.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4922.2 Δ lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4922.3 Δ lemma for ω2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4922.4 Sunflower lemma and conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4922.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

23 Suslin algebra 5023.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5023.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 51

23.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5123.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5223.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Chapter 1

Amoeba order

In mathematics, the amoeba order is the partial order of open subsets of 2ω of measure less than 1/2, ordered byreverse inclusion. Amoeba forcing is forcing with the amoeba order; it adds a measure 1 set of random reals.There are several variations, where 2ω is replaced by the real numbers or a real vector space or the unit interval, andthe number 1/2 is replaced by some positive number ε.The name “amoeba order” come from the fact that a subset in the amoeba order can “engulf” a measure zero set byextending a "pseudopod" to form a larger subset in the order containing this measure zero set, which is analogous tothe way an amoeba eats food.The amoeba order satisfies the countable chain condition.

1.1 References• Kunen, Kenneth (2011), Set theory, Studies in Logic 34, London: College Publications, ISBN 978-1-84890-050-9, MR 2905394, Zbl 1262.03001

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Chapter 2

Boolean-valued model

In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure frommodel theory. In a Boolean-valued model, the truth values of propositions are not limited to “true” and “false”, butinstead take values in some fixed complete Boolean algebra.Boolean-valued models were introduced by Dana Scott, Robert M. Solovay, and Petr Vopěnka in the 1960s in order tohelp understand Paul Cohen's method of forcing. They are also related to Heyting algebra semantics in intuitionisticlogic.

2.1 Definition

Fix a complete Boolean algebra B[1] and a first-order language L; the signature of L will consist of a collection ofconstant symbols, function symbols, and relation symbols.A Boolean-valued model for the language L consists of a universeM, which is a set of elements (or names), togetherwith interpretations for the symbols. Specifically, the model must assign to each constant symbol of L an element ofM, and to each n-ary function symbol f of L and each n-tuple <a0,...,an−₁> of elements ofM, the model must assignan element of M to the term f(a0,...,an−₁).Interpretation of the atomic formulas of L is more complicated. To each pair a and b of elements of M, the modelmust assign a truth value ||a=b|| to the expression a=b; this truth value is taken from the Boolean algebra B. Similarly,for each n-ary relation symbol R of L and each n-tuple <a0,...,an−₁> of elements of M, the model must assign anelement of B to be the truth value ||R(a0,...,an−₁)||.

2.2 Interpretation of other formulas and sentences

The truth values of the atomic formulas can be used to reconstruct the truth values of more complicated formulas,using the structure of the Boolean algebra. For propositional connectives, this is easy; one simply applies the cor-responding Boolean operators to the truth values of the subformulae. For example, if φ(x) and ψ(y,z) are formulaswith one and two free variables, respectively, and if a, b, c are elements of the model’s universe to be substituted forx, y, and z, then the truth value of

ϕ(a) ∧ ψ(b, c)

is simply

||ϕ(a) ∧ ψ(b, c)|| = ||ϕ(a)|| ∧ ||ψ(b, c)||

The completeness of the Boolean algebra is required to define truth values for quantified formulas. If φ(x) is a formulawith free variable x (and possibly other free variables that are suppressed), then

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2.3. BOOLEAN-VALUED MODELS OF SET THEORY 3

||∃xϕ(x)|| =∨a∈M

||ϕ(a)||,

where the right-hand side is to be understood as the supremum in B of the set of all truth values ||φ(a)|| as a rangesover M.The truth value of a formula is sometimes referred to as its probability. However, these are not probabilities in theordinary sense, because they are not real numbers, but rather elements of the complete Boolean algebra B.

2.3 Boolean-valued models of set theory

Given a complete Boolean algebra B[1] there is a Boolean-valued model denoted by VB, which is the Boolean-valuedanalogue of the von Neumann universe V. (Strictly speaking, VB is a proper class, so we need to reinterpret what itmeans to be a model appropriately.) Informally, the elements of VB are “Boolean-valued sets”. Given an ordinary setA, every set either is or is not a member; but given a Boolean-valued set, every set has a certain, fixed “probability” ofbeing a member of A. Again, the “probability” is an element of B, not a real number. The concept of Boolean-valuedsets resembles, but is not the same as, the notion of a fuzzy set.The (“probabilistic”) elements of the Boolean-valued set, in turn, are also Boolean-valued sets, whose elements arealso Boolean-valued sets, and so on. In order to obtain a non-circular definition of Boolean-valued set, they aredefined inductively in a hierarchy similar to the cumulative hierarchy. For each ordinal α of V, the set VBα is definedas follows.

• VB0 is the empty set.

• VBα+1 is the set of all functions from VBα to B. (Such a function represents a “probabilistic” subset of VBα;if f is such a function, then for any x∈VBα, f(x) is the probability that x is in the set.)

• If α is a limit ordinal, VBα is the union of VBβ for β<α

The class VB is defined to be the union of all sets VBα.It is also possible to relativize this entire construction to some transitive model M of ZF (or sometimes a fragmentthereof). The Boolean-valued model MB is obtained by applying the above construction inside M. The restriction totransitive models is not serious, as the Mostowski collapsing theorem implies that every “reasonable” (well-founded,extensional) model is isomorphic to a transitive one. (If the model M is not transitive things get messier, as M'sinterpretation of what it means to be a “function” or an “ordinal” may differ from the “external” interpretation.)Once the elements of VB have been defined as above, it is necessary to define B-valued relations of equality andmembership on VB. Here a B-valued relation on VB is a function from VB×VB to B. To avoid confusion with the usualequality and membership, these are denoted by ||x=y|| and ||x∈y|| for x and y in VB. They are defined as follows:

||x∈y|| is defined to be ∑t∈Dₒ ₍y₎ ||x=t|| ∧ y(t) ("x is in y if it is equal to something in y").

:||x=y|| is defined to be ||x⊆y||∧||y⊆x|| ("x equals y if x and y are both subsets of each other”), where :||x⊆y|| is definedto be ∏t∈Dₒ ₍x₎ x(t)⇒||t∈y|| ("x is a subset of y if all elements of x are in y")The symbols ∑ and ∏ denote the least upper bound and greatest lower bound operations, respectively, in the completeBoolean algebra B. At first sight the definitions above appear to be circular: || ∈ || depends on || = ||, which depends on|| ⊆ ||, which depends on || ∈ ||. However, a close examination shows that the definition of || ∈ || only depends on || ∈ ||for elements of smaller rank, so || ∈ || and || = || are well defined functions from VB×VB to B.It can be shown that the B-valued relations || ∈ || and || = || on VB make VB into a Boolean-valued model of set theory.Each sentence of first order set theory with no free variables has a truth value in B; it must be shown that the axiomsfor equality and all the axioms of ZF set theory (written without free variables) have truth value 1 (the largest elementof B). This proof is straightforward, but it is long because there are many different axioms that need to be checked.

4 CHAPTER 2. BOOLEAN-VALUED MODEL

2.4 Relationship to forcing

Set theorists use a technique called forcing to obtain independence results and to construct models of set theory forother purposes. The method was originally developed by Paul Cohen but has been greatly extended since then. Inone form, forcing “adds to the universe” a generic subset of a poset, the poset being designed to impose interestingproperties on the newly-added object. The wrinkle is that (for interesting posets) it can be proved that there simplyis no such generic subset of the poset. There are three usual ways of dealing with this:

• syntactic forcing A forcing relation p ⊩ ϕ is defined between elements p of the poset and formulas φ ofthe forcing language. This relation is defined syntactically and has no semantics; that is, no model is everproduced. Rather, starting with the assumption that ZFC (or some other axiomatization of set theory) provesthe independent statement, one shows that ZFC must also be able to prove a contradiction. However, theforcing is “over V"; that is, it is not necessary to start with a countable transitive model. See Kunen (1980) foran exposition of this method.

• countable transitive models One starts with a countable transitive model M of as much of set theory as isneeded for the desired purpose, and that contains the poset. Then there do exist filters on the poset that aregeneric over M; that is, that meet all dense open subsets of the poset that happen also to be elements of M.

• fictional generic objects Commonly, set theorists will simply pretend that the poset has a subset that is genericover all of V. This generic object, in nontrivial cases, cannot be an element of V, and therefore “does not reallyexist”. (Of course, it is a point of philosophical contention whether any sets “really exist”, but that is outside thescope of the current discussion.) Perhaps surprisingly, with a little practice this method is useful and reliable,but it can be philosophically unsatisfying.

2.4.1 Boolean-valued models and syntactic forcing

Boolean-valued models can be used to give semantics to syntactic forcing; the price paid is that the semantics is not2-valued (“true or false”), but assigns truth values from some complete Boolean algebra. Given a forcing poset P,there is a corresponding complete Boolean algebra B, often obtained as the collection of regular open subsets of P,where the topology on P is defined by declaring all lower sets open (and all upper sets closed). (Other approaches toconstructing B are discussed below.)Now the order on B (after removing the zero element) can replace P for forcing purposes, and the forcing relationcan be interpreted semantically by saying that, for p an element of B and φ a formula of the forcing language,

p ⊩ ϕ ⇐⇒ p ≤ ||ϕ||

where ||φ|| is the truth value of φ in VB.This approach succeeds in assigning a semantics to forcing over V without resorting to fictional generic objects. Thedisadvantages are that the semantics is not 2-valued, and that the combinatorics of B are often more complicated thanthose of the underlying poset P.

2.4.2 Boolean-valued models and generic objects over countable transitive models

One interpretation of forcing starts with a countable transitive model M of ZF set theory, a partially ordered set P,and a “generic” subset G of P, and constructs a new model of ZF set theory from these objects. (The conditions thatthe model be countable and transitive simplify some technical problems, but are not essential.) Cohen’s constructioncan be carried out using Boolean-valued models as follows.

• Construct a complete Boolean algebra B as the complete Boolean algebra “generated by” the poset P.

• Construct an ultrafilter U on B (or equivalently a homomorphism from B to the Boolean algebra {true, false})from the generic subset G of P.

• Use the homomorphism from B to {true, false} to turn the Boolean-valued modelMB of the section above intoan ordinary model of ZF.

2.5. NOTES 5

We now explain these steps in more detail.For any poset P there is a complete Boolean algebra B and a map e from P to B+ (the non-zero elements of B) suchthat the image is dense, e(p)≤e(q) whenever p≤q, and e(p)e(q)=0 whenever p and q are incompatible. This Booleanalgebra is unique up to isomorphism. It can be constructed as the algebra of regular open sets in the topological spaceof P (with underlying set P, and a base given by the sets Up of elements q with q≤p).The map from the poset P to the complete Boolean algebra B is not injective in general. The map is injective if andonly if P has the following property: if every r≤p is compatible with q, then p≤q.The ultrafilter U on B is defined to be the set of elements b of B that are greater than some element of (the imageof) G. Given an ultrafilter U on a Boolean algebra, we get a homomorphism to {true, false} by mapping U to trueand its complement to false. Conversely, given such a homomorphism, the inverse image of true is an ultrafilter, soultrafilters are essentially the same as homomorphisms to {true, false}. (Algebraists might prefer to use maximalideals instead of ultrafilters: the complement of an ultrafilter is a maximal ideal, and conversely the complement of amaximal ideal is an ultrafilter.)If g is a homomorphism from a Boolean algebra B to a Boolean algebra C and MB is any B-valued model of ZF (orof any other theory for that matter) we can turn MB into a C -valued model by applying the homomorphism g tothe value of all formulas. In particular if C is {true, false} we get a {true, false}-valued model. This is almost thesame as an ordinary model: in fact we get an ordinary model on the set of equivalence classes under || = || of a {true,false}-valued model. So we get an ordinary model of ZF set theory by starting from M, a Boolean algebra B, andan ultrafilter U on B. (The model of ZF constructed like this is not transitive. In practice one applies the Mostowskicollapsing theorem to turn this into a transitive model.)We have seen that forcing can be done using Boolean-valuedmodels, by constructing a Boolean algebra with ultrafilterfrom a poset with a generic subset. It is also possible to go back the other way: given a Boolean algebra B, we canform a poset P of all the nonzero elements of B, and a generic ultrafilter on B restricts to a generic set on P. So thetechniques of forcing and Boolean-valued models are essentially equivalent.

2.5 Notes[1] B here is assumed to be nondegenerate; that is, 0 and 1 must be distinct elements of B. Authors writing on Boolean-valued

models typically take this requirement to be part of the definition of “Boolean algebra”, but authors writing on Booleanalgebras in general often do not.

2.6 References• Bell, J. L. (1985) Boolean-ValuedModels and Independence Proofs in Set Theory, Oxford. ISBN 0-19-853241-5

• Grishin, V.N. (2001), “b/b016990”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2. OCLC 174929965.

• Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. North-Holland. ISBN 0-444-85401-0. OCLC 12808956.

• Kusraev, A. G. and S. S. Kutateladze (1999). Boolean Valued Analysis. Kluwer Academic Publishers. ISBN0-7923-5921-6. OCLC 41967176. Contains an account of Boolean-valued models and applications to Rieszspaces, Banach spaces and algebras.

• Manin, Yu. I. (1977). A Course in Mathematical Logic. Springer. ISBN 0-387-90243-0. OCLC 2797938.Contains an account of forcing and Boolean-valuedmodels written formathematicians who are not set theorists.

• Rosser, J. Barkley (1969). Simply Independence Proofs, Boolean valued models of set theory. Academic Press.

Chapter 3

Cantor algebra

For the algebras encoding a bijection from an infinite set X onto the product X×X, sometimes called Cantor algebras,see Jónsson–Tarski algebra.

In mathematics, a Cantor algebra, named after Georg Cantor, is one of two closely related Boolean algebras, onecountable and one complete.The countable Cantor algebra is the Boolean algebra of all clopen subsets of the Cantor set. This is the free Booleanalgebra on a countable number of generators. Up to isomorphism, this is the only nontrivial Boolean algebra that isboth countable and atomless.The complete Cantor algebra is the complete Boolean algebra of Borel subsets of the reals modulo meager sets (Balcar& Jech 2006). It is isomorphic to the completion of the countable Cantor algebra. (The complete Cantor algebra issometimes called the Cohen algebra, though "Cohen algebra" usually refers to a different type of Boolean algebra.)The complete Cantor algebra was studied by von Neumann in 1935 (later published as (von Neumann 1998)), whoshowed that it is not isomorphic to the random algebra of Borel subsets modulo measure zero sets.

3.1 References• Balcar, Bohuslav; Jech, Thomas (2006), “Weak distributivity, a problem of von Neumann and the mystery ofmeasurability”, Bulletin of Symbolic Logic 12 (2): 241–266, MR 2223923

• von Neumann, John (1998) [1960], Continuous geometry, Princeton Landmarks in Mathematics, PrincetonUniversity Press, ISBN 978-0-691-05893-1, MR 0120174

6

Chapter 4

Cohen algebra

Not to be confused with Cohen ring or Rankin–Cohen algebra.For the quotient of the algebra of Borel sets by the ideal of meager sets, sometimes called the Cohen algebra, seeCantor algebra.

In mathematical set theory, a Cohen algebra, named after Paul Cohen, is a type of Boolean algebra used in thetheory of forcing. A Cohen algebra is a Boolean algebra whose completion is isomorphic to the completion of a freeBoolean algebra (Koppelberg 1993).

4.1 References• Koppelberg, Sabine (1993), “Characterizations of Cohen algebras”, Papers on general topology and applications

(Madison, WI, 1991), Annals of the New York Academy of Sciences 704, New York Academy of Sciences,pp. 222–237, doi:10.1111/j.1749-6632.1993.tb52525.x, MR 1277859

7

Chapter 5

Collapsing algebra

In mathematics, a collapsing algebra is a type of Boolean algebra sometimes used in forcing to reduce (“collapse”)the size of cardinals. The posets used to generate collapsing algebras were introduced by Azriel Lévy (1963).The collapsing algebra of λω is a complete Boolean algebra with at least λ elements but generated by a countablenumber of elements. As the size of countably generated complete Boolean algebras is unbounded, this shows thatthere is no free complete Boolean algebra on a countable number of elements.

5.1 Definition

There are several slightly different sorts of collapsing algebras.If κ and λ are cardinals, then the Boolean algebra of regular open sets of the product space κλ is a collapsing algebra.Here κ and λ are both given the discrete topology. There are several different options for the topology of κλ. Thesimplest option is to take the usual product topology. Another option is to take the topology generated by open setsconsisting of functions whose value is specified on less than λ elements of λ.

5.2 References• Bell, J. L. (1985). Boolean-Valued Models and Independence Proofs in Set Theory. Oxford Logic Guides 12(2nd ed.). Oxford: Oxford University Press (Clarendon Press). ISBN 0-19-853241-5. Zbl 0585.03021.

• Jech, Thomas (2003). Set theory (third millennium (revised and expanded) ed.). Springer-Verlag. ISBN 3-540-44085-2. OCLC 174929965. Zbl 1007.03002.

• Lévy, Azriel (1963). “Independence results in set theory by Cohen’s method. IV,”. Notices Amer. Math. Soc.10: 593.

8

Chapter 6

Complete Boolean algebra

This article is about a type of mathematical structure. For complete sets of Boolean operators, see Functional com-pleteness.

In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (leastupper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theoryof forcing. Every Boolean algebra A has an essentially unique completion, which is a complete Boolean algebracontaining A such that every element is the supremum of some subset of A. As a partially ordered set, this completionof A is the Dedekind–MacNeille completion.More generally, if κ is a cardinal then a Boolean algebra is called κ-complete if every subset of cardinality less thanκ has a supremum.

6.1 Examples• Every finite Boolean algebra is complete.

• The algebra of subsets of a given set is a complete Boolean algebra.

• The regular open sets of any topological space form a complete Boolean algebra. This example is of particularimportance because every forcing poset can be considered as a topological space (a base for the topologyconsisting of sets that are the set of all elements less than or equal to a given element). The correspondingregular open algebra can be used to formBoolean-valuedmodels which are then equivalent to generic extensionsby the given forcing poset.

• The algebra of all measurable subsets of a σ-finite measure space, modulo null sets, is a complete Booleanalgebra. When the measure space is the unit interval with the σ-algebra of Lebesgue measurable sets, theBoolean algebra is called the random algebra.

• The algebra of all measurable subsets of a measure space is a ℵ1-complete Boolean algebra, but is not usuallycomplete.

• The algebra of all subsets of an infinite set that are finite or have finite complement is a Boolean algebra but isnot complete.

• The Boolean algebra of all Baire sets modulo meager sets in a topological space with a countable base iscomplete; when the topological space is the real numbers the algebra is sometimes called the Cantor algebra.

• Another example of a Boolean algebra that is not complete is the Boolean algebra P(ω) of all sets of naturalnumbers, quotiented out by the ideal Fin of finite subsets. The resulting object, denoted P(ω)/Fin, consists ofall equivalence classes of sets of naturals, where the relevant equivalence relation is that two sets of naturals are

9

10 CHAPTER 6. COMPLETE BOOLEAN ALGEBRA

equivalent if their symmetric difference is finite. The Boolean operations are defined analogously, for example,if A and B are two equivalence classes in P(ω)/Fin, we defineA∧B to be the equivalence class of a∩b , wherea and b are some (any) elements of A and B respectively.

Now let a0, a1,... be pairwise disjoint infinite sets of naturals, and let A0, A1,... be their correspondingequivalence classes in P(ω)/Fin . Then given any upper bound X of A0, A1,... in P(ω)/Fin, we can finda lesser upper bound, by removing from a representative for X one element of each an. Therefore theAn have no supremum.

• A Boolean algebra is complete if and only if its Stone space of prime ideals is extremally disconnected.

6.2 Properties of complete Boolean algebras

• Sikorski’s extension theorem states that

if A is a subalgebra of a Boolean algebra B, then any homomorphism from A to a complete Boolean algebra C can beextended to a morphism from B to C.

• Every subset of a complete Boolean algebra has a supremum, by definition; it follows that every subset also hasan infimum (greatest lower bound).

• For a complete boolean algebra both infinite distributive laws hold.

• For a complete boolean algebra infinite de-Morgan’s laws hold.

6.3 The completion of a Boolean algebra

The completion of a Boolean algebra can be defined in several equivalent ways:

• The completion of A is (up to isomorphism) the unique complete Boolean algebra B containing A such that Ais dense in B; this means that for every nonzero element of B there is a smaller non-zero element of A.

• The completion of A is (up to isomorphism) the unique complete Boolean algebra B containing A such thatevery element of B is the supremum of some subset of A.

The completion of a Boolean algebra A can be constructed in several ways:

• The completion is the Boolean algebra of regular open sets in the Stone space of prime ideals of A. Eachelement x of A corresponds to the open set of prime ideals not containing x (which open and closed, andtherefore regular).

• The completion is the Boolean algebra of regular cuts of A. Here a cut is a subset U of A+ (the non-zeroelements of A) such that if q is in U and p≤q then p is in U, and is called regular if whenever p is not in U thereis some r ≤ p such that U has no elements ≤r. Each element p of A corresponds to the cut of elements ≤p.

If A is a metric space and B its completion then any isometry from A to a complete metric space C can be extended toa unique isometry from B to C. The analogous statement for complete Boolean algebras is not true: a homomorphismfrom a Boolean algebraA to a complete Boolean algebra C cannot necessarily be extended to a (supremum preserving)homomorphism of complete Boolean algebras from the completion B of A to C. (By Sikorski’s extension theorem itcan be extended to a homomorphism of Boolean algebras from B to C, but this will not in general be a homomorphismof complete Boolean algebras; in other words, it need not preserve suprema.)

6.4. FREE Κ-COMPLETE BOOLEAN ALGEBRAS 11

6.4 Free κ-complete Boolean algebras

Unless the Axiom of Choice is relaxed,[1] free complete boolean algebras generated by a set do not exist (unless the setis finite). More precisely, for any cardinal κ, there is a complete Boolean algebra of cardinality 2κ greater than κ thatis generated as a complete Boolean algebra by a countable subset; for example the Boolean algebra of regular opensets in the product space κω, where κ has the discrete topology. A countable generating set consists of all sets am,nfor m, n integers, consisting of the elements x∈κω such that x(m)<x(n). (This boolean algebra is called a collapsingalgebra, because forcing with it collapses the cardinal κ onto ω.)In particular the forgetful functor from complete Boolean algebras to sets has no left adjoint, even though it is contin-uous and the category of Boolean algebras is small-complete. This shows that the “solution set condition” in Freyd’sadjoint functor theorem is necessary.Given a set X, one can form the free Boolean algebra A generated by this set and then take its completion B. HoweverB is not a “free” complete Boolean algebra generated by X (unless X is finite or AC is omitted), because a functionfrom X to a free Boolean algebra C cannot in general be extended to a (supremum-preserving) morphism of Booleanalgebras from B to C.On the other hand, for any fixed cardinal κ, there is a free (or universal) κ-complete Boolean algebra generated byany given set.

6.5 See also• Complete lattice

• Complete Heyting algebra

6.6 References[1] Stavi, Jonathan (1974), “A model of ZF with an infinite free complete Boolean algebra” (reprint), Israel Journal of Math-

ematics 20 (2): 149–163, doi:10.1007/BF02757883.

• Johnstone, Peter T. (1982), Stone spaces, Cambridge University Press, ISBN 0-521-33779-8

• Koppelberg, Sabine (1989), Monk, J. Donald; Bonnet, Robert, eds., Handbook of Boolean algebras 1, Ams-terdam: North-Holland Publishing Co., pp. xx+312, ISBN 0-444-70261-X, MR 0991565

• Monk, J. Donald; Bonnet, Robert, eds. (1989), Handbook of Boolean algebras 2, Amsterdam: North-HollandPublishing Co., ISBN 0-444-87152-7, MR 0991595

• Monk, J. Donald; Bonnet, Robert, eds. (1989), Handbook of Boolean algebras 3, Amsterdam: North-HollandPublishing Co., ISBN 0-444-87153-5, MR 0991607

• Vladimirov, D.A. (2001), “Boolean algebra”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

Chapter 7

Continuum hypothesis

This article is about the hypothesis in set theory. For the assumption in fluid mechanics, see Fluid mechanics.

In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states:

There is no set whose cardinality is strictly between that of the integers and the real numbers.

The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the firstof Hilbert’s 23 problems presented in the year 1900. Τhe answer to this problem is independent of ZFC set theory(that is, Zermelo–Fraenkel set theory with the axiom of choice included), so that either the continuum hypothesis orits negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFCis consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in1940.The name of the hypothesis comes from the term the continuum for the real numbers. It is abbreviated CH.

7.1 Cardinality of infinite sets

Main article: Cardinal number

Two sets are said to have the same cardinality or cardinal number if there exists a bijection (a one-to-one correspon-dence) between them. Intuitively, for two sets S and T to have the same cardinality means that it is possible to “pairoff” elements of S with elements of T in such a fashion that every element of S is paired off with exactly one elementof T and vice versa. Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green} .With infinite sets such as the set of integers or rational numbers, this becomes more complicated to demonstrate.The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a propersubset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rationalnumbers than integers, and more real numbers than rational numbers. However, this intuitive analysis does not takeaccount of the fact that all three sets are infinite. It turns out the rational numbers can actually be placed in one-to-onecorrespondence with the integers, and therefore the set of rational numbers is the same size (cardinality) as the set ofintegers: they are both countable sets.Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers(see Cantor’s first uncountability proof and Cantor’s diagonal argument). His proofs, however, give no indication ofthe extent to which the cardinality of the integers is less than that of the real numbers. Cantor proposed the continuumhypothesis as a possible solution to this question.The hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinalityof the set of integers. Equivalently, as the cardinality of the integers is ℵ0 ("aleph-naught") and the cardinality of thereal numbers is 2ℵ0 (i.e. it equals the cardinality of the power set of the integers), the continuum hypothesis says thatthere is no set S for which

12

7.2. INDEPENDENCE FROM ZFC 13

ℵ0 < |S| < 2ℵ0 .

Assuming the axiom of choice, there is a smallest cardinal number ℵ1 greater than ℵ0 , and the continuum hypothesisis in turn equivalent to the equality

2ℵ0 = ℵ1.

A consequence of the continuum hypothesis is that every infinite subset of the real numbers either has the samecardinality as the integers or the same cardinality as the entire set of the reals.There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis (GCH)which says that for all ordinals α

2ℵα = ℵα+1.

That is, GCH asserts that the cardinality of the power set of any infinite set is the smallest cardinality greater thanthat of the set.

7.2 Independence from ZFC

Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain (Dauben 1990). Itbecame the first on David Hilbert’s list of important open questions that was presented at the International Congressof Mathematicians in the year 1900 in Paris. Axiomatic set theory was at that point not yet formulated.Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standardZermelo–Fraenkel set theory (ZF), even if the axiom of choice is adopted (ZFC) (Gödel (1940)). Paul Cohen showedin 1963 that CH cannot be proven from those same axioms either (Cohen (1963) & Cohen (1964)). Hence, CH isindependent of ZFC. Both of these results assume that the Zermelo–Fraenkel axioms are consistent; this assumptionis widely believed to be true. Cohen was awarded the Fields Medal in 1966 for his proof.The continuum hypothesis is closely related to many statements in analysis, point set topology and measure theory.As a result of its independence, many substantial conjectures in those fields have subsequently been shown to beindependent as well.So far, CH appears to be independent of all known large cardinal axioms in the context of ZFC. (Feferman (1999))The independence from ZFC means that proving or disproving the CH within ZFC is impossible. However, Gödeland Cohen’s negative results are not universally accepted as disposing of the hypothesis. Hilbert’s problem remainsan active topic of research; see Woodin (2001) and Koellner (2011a) for an overview of the current research status.The continuum hypothesis was not the first statement shown to be independent of ZFC. An immediate consequenceof Gödel’s incompleteness theorem, which was published in 1931, is that there is a formal statement (one for eachappropriate Gödel numbering scheme) expressing the consistency of ZFC that is independent of ZFC, assuming thatZFC is consistent. The continuum hypothesis and the axiom of choice were among the first mathematical statementsshown to be independent of ZF set theory. These proofs of independence were not completed until Paul Cohendeveloped forcing in the 1960s. They all rely on the assumption that ZF is consistent. These proofs are called proofsof relative consistency (see Forcing (mathematics)).A result of Solovay, proved shortly after Cohen’s result on the independence of the continuum hypothesis, shows thatin any model of ZFC, if κ is a cardinal of uncountable cofinality, then there is a forcing extension in which 2ℵ0 = κ. However, it is not consistent to assume 2ℵ0 is ℵω or ℵω1+ω or any cardinal with cofinality ω .

7.3 Arguments for and against CH

Gödel believed that CH is false and that his proof that CH is consistent with ZFC only shows that the Zermelo–Fraenkel axioms do not adequately characterize the universe of sets. Gödel was a platonist and therefore had no

14 CHAPTER 7. CONTINUUM HYPOTHESIS

problems with asserting the truth and falsehood of statements independent of their provability. Cohen, though aformalist (Goodman 1979), also tended towards rejecting CH.Historically, mathematicians who favored a “rich” and “large” universe of sets were against CH, while those favoringa “neat” and “controllable” universe favored CH. Parallel arguments were made for and against the axiom of con-structibility, which implies CH. More recently, Matthew Foreman has pointed out that ontological maximalism canactually be used to argue in favor of CH, because among models that have the same reals, models with “more” setsof reals have a better chance of satisfying CH (Maddy 1988, p. 500).Another viewpoint is that the conception of set is not specific enough to determine whether CH is true or false. Thisviewpoint was advanced as early as 1923 by Skolem, even beforeGödel’s first incompleteness theorem. Skolem arguedon the basis of what is now known as Skolem’s paradox, and it was later supported by the independence of CH fromthe axioms of ZFC, since these axioms are enough to establish the elementary properties of sets and cardinalities. Inorder to argue against this viewpoint, it would be sufficient to demonstrate new axioms that are supported by intuitionand resolve CH in one direction or another. Although the axiom of constructibility does resolve CH, it is not generallyconsidered to be intuitively true any more than CH is generally considered to be false (Kunen 1980, p. 171).At least two other axioms have been proposed that have implications for the continuum hypothesis, although theseaxioms have not currently found wide acceptance in the mathematical community. In 1986, Chris Freiling presentedan argument against CH by showing that the negation of CH is equivalent to Freiling’s axiom of symmetry, a statementabout probabilities. Freiling believes this axiom is “intuitively true” but others have disagreed. A difficult argumentagainst CH developed by W. Hugh Woodin has attracted considerable attention since the year 2000 (Woodin 2001a,2001b). Foreman (2003) does not reject Woodin’s argument outright but urges caution.Solomon Feferman (2011) has made a complex philosophical argument that CH is not a definite mathematical prob-lem. He proposes a theory of “definiteness” using a semi-intuitionistic subsystem of ZF that accepts classical logic forbounded quantifiers but uses intuitionistic logic for unbounded ones, and suggests that a proposition ϕ is mathemati-cally “definite” if the semi-intuitionistic theory can prove (ϕ∨¬ϕ) . He conjectures that CH is not definite accordingto this notion, and proposes that CH should therefore be considered not to have a truth value. Peter Koellner (2011b)wrote a critical commentary on Feferman’s article.Joel David Hamkins proposes a multiverse approach to set theory and argues that “the continuum hypothesis is settledon the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can nolonger be settled in the manner formerly hoped for.” (Hamkins 2012). In a related vein, Saharon Shelah wrote thathe does “not agree with the pure Platonic view that the interesting problems in set theory can be decided, that we justhave to discover the additional axiom. My mental picture is that we have many possible set theories, all conformingto ZFC.” (Shelah 2003).

7.4 The generalized continuum hypothesis

The generalized continuum hypothesis (GCH) states that if an infinite set’s cardinality lies between that of an infiniteset S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as thepower set of S. That is, for any infinite cardinal λ there is no cardinal κ such that λ < κ < 2λ. GCH is equivalentto:

ℵα+1 = 2ℵα for every ordinal α. (occasionally called Cantor’s aleph hypothesis)

The beth numbers provide an alternate notation for this condition: ℵα = ℶα for every ordinal α.This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power setof the integers. It was first suggested by Jourdain (1905).Like CH, GCH is also independent of ZFC, but Sierpiński proved that ZF + GCH implies the axiom of choice (AC),so choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. Toprove this, Sierpiński showed GCH implies that every cardinality n is smaller than some Aleph number, and thus canbe ordered. This is done by showing that n is smaller than 2ℵ0+n which is smaller than its own Hartogs number —this uses the equality 2ℵ0+n = 2 · 2ℵ0+n ; for the full proof, see Gillman (2002).Kurt Gödel showed that GCH is a consequence of ZF + V=L (the axiom that every set is constructible relative to theordinals), and is therefore consistent with ZFC. As GCH implies CH, Cohen’s model in which CH fails is a modelin which GCH fails, and thus GCH is not provable from ZFC. W. B. Easton used the method of forcing developed

7.5. SEE ALSO 15

by Cohen to prove Easton’s theorem, which shows it is consistent with ZFC for arbitrarily large cardinals ℵα tofail to satisfy 2ℵα = ℵα+1. Much later, Foreman and Woodin proved that (assuming the consistency of very largecardinals) it is consistent that 2κ > κ+ holds for every infinite cardinal κ. Later Woodin extended this by showingthe consistency of 2κ = κ++ for every κ . A recent result of Carmi Merimovich shows that, for each n≥1, it isconsistent with ZFC that for each κ, 2κ is the nth successor of κ. On the other hand, László Patai (1930) proved, thatif γ is an ordinal and for each infinite cardinal κ, 2κ is the γth successor of κ, then γ is finite.For any infinite sets A and B, if there is an injection from A to B then there is an injection from subsets of A to subsetsof B. Thus for any infinite cardinals A and B,

A < B → 2A ≤ 2B.

If A and B are finite, the stronger inequality

A < B → 2A < 2B

holds. GCH implies that this strict, stronger inequality holds for infinite cardinals as well as finite cardinals.

7.4.1 Implications of GCH for cardinal exponentiation

Although the generalized continuum hypothesis refers directly only to cardinal exponentiation with 2 as the base, onecan deduce from it the values of cardinal exponentiation in all cases. It implies that ℵℵβ

α is (see: Hayden & Kennison(1968), page 147, exercise 76):

ℵβ+1 when α ≤ β+1;ℵα when β+1 < α and ℵβ < cf(ℵα) where cf is the cofinality operation; andℵα+1 when β+1 < α and ℵβ ≥ cf(ℵα) .

7.5 See also

• Aleph number

• Beth number

• Cardinality

• Ω-logic

• Wetzel’s problem

7.6 References

• Cohen, Paul Joseph (2008) [1966]. Set theory and the continuum hypothesis. Mineola, New York: DoverPublications. ISBN 978-0-486-46921-8.

• Cohen, Paul J. (December 15, 1963). “The Independence of the Continuum Hypothesis”. Proceedings of theNational Academy of Sciences of the United States of America 50 (6): 1143–1148. doi:10.1073/pnas.50.6.1143.JSTOR 71858. PMC 221287. PMID 16578557.

• Cohen, Paul J. (January 15, 1964). “The Independence of the Continuum Hypothesis, II”. Proceedings of theNational Academy of Sciences of the United States of America 51 (1): 105–110. doi:10.1073/pnas.51.1.105.JSTOR 72252. PMC 300611. PMID 16591132.

• Dales, H. G.; Woodin, W. H. (1987). An Introduction to Independence for Analysts. Cambridge.

16 CHAPTER 7. CONTINUUM HYPOTHESIS

• Dauben, Joseph Warren (1990). Georg Cantor: His Mathematics and Philosophy of the Infinite. PrincetonUniversity Press. pp. 134–137. ISBN 9780691024479.

• Enderton, Herbert (1977). Elements of Set Theory. Academic Press.

• Feferman, Solomon (February 1999). “Doesmathematics need new axioms?". AmericanMathematicalMonthly106 (2): 99–111. doi:10.2307/2589047.

• Feferman, Solomon (2011). “Is the ContinuumHypothesis a definitemathematical problem?" (PDF).Exploringthe Frontiers of Independence (Harvard lecture series). External link in |work= (help)

• Foreman, Matt (2003). “Has the Continuum Hypothesis been Settled?" (PDF). Retrieved February 25, 2006.

• Freiling, Chris (1986). “Axioms of Symmetry: Throwing Darts at the Real Number Line”. Journal of SymbolicLogic (Association for Symbolic Logic) 51 (1): 190–200. doi:10.2307/2273955. JSTOR 2273955.

• Gödel, K. (1940). The Consistency of the Continuum-Hypothesis. Princeton University Press.

• Gillman, Leonard (2002). “Two Classical Surprises Concerning the Axiom of Choice and the ContinuumHypothesis” (PDF). American Mathematical Monthly 109. doi:10.2307/2695444.

• Gödel, K.: What is Cantor’s Continuum Problem?, reprinted in Benacerraf and Putnam’s collection Philosophyof Mathematics, 2nd ed., Cambridge University Press, 1983. An outline of Gödel’s arguments against CH.

• Goodman, Nicolas D. (1979). “Mathematics as an objective science”. The American Mathematical Monthly86 (7): 540–551. doi:10.2307/2320581. MR 542765. This view is often called formalism. Positions more orless like this may be found in Haskell Curry [5], Abraham Robinson [17], and Paul Cohen [4].

• Joel David Hamkins. The set-theoretic multiverse. Rev. Symb. Log. 5 (2012), no. 3, 416–449.

• Seymour Hayden and John F. Kennison: Zermelo–Fraenkel Set Theory (1968), Charles E. Merrill PublishingCompany, Columbus, Ohio.

• Jourdain, Philip E. B. (1905). “On transfinite cardinal numbers of the exponential form”. Philosophical Mag-azine, Series 6 9: 42–56. doi:10.1080/14786440509463254.

• Koellner, Peter (2011a). “The Continuum Hypothesis” (PDF). Exploring the Frontiers of Independence (Har-vard lecture series).

• Koellner, Peter (2011b). “Feferman On the Indefiniteness of CH” (PDF).

• Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Amsterdam: North-Holland.ISBN 978-0-444-85401-8.

• Maddy, Penelope (June 1988). “Believing the Axioms, I”. Journal of Symbolic Logic (Association for SymbolicLogic) 53 (2): 481–511. doi:10.2307/2274520. JSTOR 2274520.

• Martin, D. (1976). “Hilbert’s first problem: the continuum hypothesis,” inMathematical Developments ArisingfromHilbert’s Problems, Proceedings of Symposia in PureMathematics XXVIII, F. Browder, editor. AmericanMathematical Society, 1976, pp. 81–92. ISBN 0-8218-1428-1

• McGough, Nancy. “The Continuum Hypothesis”.

• Merimovich, Carmi (2007). “A power function with a fixed finite gap everywhere”. Journal of Symbolic Logic72 (2): 361–417. doi:10.2178/jsl/1185803615.

• Moore, Gregory H. (2011). “Early history of the generalized continuum hypothesis: 1878–1938”. Bull. Sym-bolic Logic 17 (4): 489–532. doi:10.2178/bsl/1318855631. MR 2896574.

• Shelah, Saharon (2003). “Logical dreams”. Bull. Amer. Math. Soc. (N.S.) 40 (2): 203–228. doi:10.1090/s0273-0979-03-00981-9.

• Woodin, W. Hugh (2001a). “The Continuum Hypothesis, Part I” (PDF). Notices of the AMS 48 (6): 567–576.

• Woodin, W. Hugh (2001b). “The ContinuumHypothesis, Part II” (PDF).Notices of the AMS 48 (7): 681–690.

7.7. EXTERNAL LINKS 17

German literature

• Cantor, Georg (1878). “Ein Beitrag zur Mannigfaltigkeitslehre”. Journal für die Reine und Angewandte Math-ematik 84: 242–258. doi:10.1515/crll.1878.84.242.

• Patai, L. (1930). “Untersuchungen über die .”reihe-א Mathematische und naturwissenschaftliche Berichte ausUngarn 37: 127–142.

7.7 External links• Szudzik, Matthew and Weisstein, Eric W., “Continuum Hypothesis”, MathWorld.

This article incorporates material from Generalized continuum hypothesis on PlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

Chapter 8

Countable chain condition

See also: Forcing (set theory) § The countable chain condition

In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strongantichain in X is countable. There are really two conditions: the upwards and downwards countable chain conditions.These are not equivalent. The countable chain condition means the downwards countable chain condition, in otherwords no two elements have a common lower bound.This is called the “countable chain condition” rather than the more logical term “countable antichain condition” forhistorical reasons related to certain chains of open sets in topological spaces and chains in complete Boolean algebras,where chain conditions sometimes happen to be equivalent to antichain conditions. For example, if κ is a cardinal,then in a complete Boolean algebra every antichain has size less than κ if and only if there is no descending κ-sequenceof elements, so chain conditions are equivalent to antichain conditions.Partial orders and spaces satisfying the ccc are used in the statement of Martin’s axiom.In the theory of forcing, ccc partial orders are used because forcing with any generic set over such an order preservescardinals and cofinalities. Furthermore, the c.c.c. property is preserved by finite support iterations (see iteratedforcing).More generally, if κ is a cardinal then a poset is said to satisfy the κ-chain condition if every antichain has size lessthan κ. The countable chain condition is the ℵ1-chain condition.

8.1 Examples and properties in topology

A topological space is said to satisfy the countable chain condition, or Suslin’s Condition, if the partially orderedset of non-empty open subsets of X satisfies the countable chain condition, i.e. every pairwise disjoint collection ofnon-empty open subsets of X is countable. The name originates from Suslin’s Problem.

• Every separable topological space is ccc. Furthermore, the product space of at most c separable spaces is aseparable space and, thus, ccc.

• Every metric space is ccc if and only if it’s separable, but in general a ccc topological space need not beseparable.

For example,

{0, 1}22ℵ0

with the product topology is ccc but not separable.

18

8.2. REFERENCES 19

8.2 References• Jech, Thomas (2003), Set Theory: Millennium Edition, Springer Monographs in Mathematics, Berlin, NewYork: Springer-Verlag, ISBN 978-3-540-44085-7

• Products of Separable Spaces, K. A. Ross, and A. H. Stone. The American Mathematical Monthly 71(4):pp.398–403 (1964)

Chapter 9

Easton’s theorem

In set theory, Easton’s theorem is a result on the possible cardinal numbers of powersets. Easton (1970) (extendinga result of Robert M. Solovay) showed via forcing that

κ < cf(2κ)

and, for κ < λ , that

2κ ≤ 2λ

are the only constraints on permissible values for 2κ when κ is a regular cardinal.

9.1 Statement of the theorem

Easton’s theorem states that if G is a class function whose domain consists of ordinals and whose range consists ofordinals such that

1. G is non-decreasing,

2. the cofinality of ℵG(α) is greater than ℵα for each α in the domain of G, and

3. ℵα is regular for each α in the domain of G,

then there is a model of ZFC such that

2ℵα = ℵG(α)

for each α in the domain of G.The proof of Easton’s theorem uses forcing with a proper class of forcing conditions over a model satisfying thegeneralized continuum hypothesis.The first two conditions in the theorem are necessary. Condition 1 is a well known property of cardinality, whilecondition 2 follows from König’s theorem.In Easton’s model the powersets of singular cardinals have the smallest possible cardinality compatible with theconditions that 2κ has cofinality greater than κ and is a non-decreasing function of κ.

20

9.2. NO EXTENSION TO SINGULAR CARDINALS 21

9.2 No extension to singular cardinals

Silver (1975) proved that a singular cardinal of uncountable cofinality cannot be the smallest cardinal for which thegeneralized continuum hypothesis fails. This shows that Easton’s theorem cannot be extended to the class of allcardinals. The program of PCF theory gives results on the possible values of 2λ for singular cardinals λ . PCF theoryshows that the values of the continuum function on singular cardinals are strongly influenced by the values on smallercardinals, whereas Easton’s theorem shows that the values of the continuum function on regular cardinals are onlyweakly influenced by the values on smaller cardinals.

9.3 See also• Singular cardinal hypothesis

• Aleph number

• Beth number

9.4 References• Easton, W. (1970), “Powers of regular cardinals”, Ann. Math. Logic 1 (2): 139–178, doi:10.1016/0003-4843(70)90012-4

• Silver, Jack (1975), “On the singular cardinals problem”, Proceedings of the International Congress of Mathe-maticians (Vancouver, B. C., 1974) 1, Montreal, Que.: Canad. Math. Congress, pp. 265–268, MR 0429564

Chapter 10

Forcing (mathematics)

For the use of forcing in recursion theory, see Forcing (recursion theory).

In the mathematical discipline of set theory, forcing is a technique discovered by Paul Cohen for proving consistencyand independence results. It was first used, in 1963, to prove the independence of the axiom of choice and thecontinuum hypothesis from Zermelo–Fraenkel set theory. Forcing was considerably reworked and simplified in thefollowing years, and has since served as a powerful technique both in set theory and in areas of mathematical logicsuch as recursion theory.Descriptive set theory uses the notion of forcing from both recursion theory and set theory. Forcing has also beenused in model theory but it is common in model theory to define genericity directly without mention of forcing.

10.1 Intuitions

Forcing is equivalent to the method of Boolean-valued models, which some feel is conceptually more natural andintuitive, but usually much more difficult to apply.Intuitively, forcing consists of expanding the set theoretical universeV to a larger universeV*. In this bigger universe,for example, one might have lots of new subsets ofω = {0,1,2,…} that were not there in the old universe, and therebyviolate the continuum hypothesis. While impossible on the face of it, this is just another version of Cantor’s paradoxabout infinity. In principle, one could consider

V ∗ = V × {0, 1},

identify x ∈ V with (x, 0) , and then introduce an expanded membership relation involving the “new” sets of theform (x, 1) . Forcing is a more elaborate version of this idea, reducing the expansion to the existence of one new set,and allowing for fine control over the properties of the expanded universe.Cohen’s original technique, now called ramified forcing, is slightly different from the unramified forcing expoundedhere.

10.2 Forcing posets

A forcing poset is an ordered triple, (P, ≤, 1), where ≤ is a preorder on P that satisfies following splitting condition:

• For all p ∈ P, there are q, r ∈ P such that q, r ≤ p with no s ∈ P such that s ≤ q, r

The largest element of P is 1, that is, p ≤ 1 for all p ∈ P.Members of P are called forcing conditions or just conditions.One reads p ≤ q as p is stronger than q. Intuitively, the “smaller” condition provides “more” information, just as thesmaller interval [3.1415926,3.1415927] provides more information about the number π than the interval [3.1,3.2]does.

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10.2. FORCING POSETS 23

There are various conventions in use. Some authors require ≤ to also be antisymmetric, so that the relation is a partialorder. Some use the term partial order anyway, conflicting with standard terminology, while some use the termpreorder. The largest element can be dispensed with. The reverse ordering is also used, most notably by SaharonShelah and his co-authors.

10.2.1 P-names

Associated with a forcing poset P is the class V(P) of P-names. P-names are sets of the form

• {(u, p) : u is a P-name and p ∈ P and (some criterion involving u and p)}

Using transfinite recursion, one defines

• Name(0) = {} ,

• Name(α + 1) = the power set of (Name(α) × P),

• Name(λ) = ∪{Name(α) : α < λ for λ a limit ordinal} ,

and then the class of P-names is defined by

V(P) = ∪{Name(α) : α is an ordinal} .

The P-names are, in fact, an expansion of the universe. Given x ∈ V, one defines xˇ to be the P-name

xˇ = {(yˇ, 1) : y ∈ x} .

Again, this is really a definition by transfinite recursion.

10.2.2 Interpretation

Given any subset G of P, one next defines the interpretation or valuation map from P-names by

val(u, G) = {val(v, G) : ∃ p ∈ G , (v, p) ∈ u} .

(Again a definition by transfinite recursion.) Note that if 1 is in G, then

val(xˇ, G) = x.

One defines

G = {(pˇ, p) : p ∈ G} ,

so that

val(G,G) = G.

10.2.3 Example

A good example of a forcing poset is (Bor(I) , ⊆ , I ) where I = [0,1] and Bor(I) are the Borel subsets of I havingnon-zero Lebesgue measure. In this case, one can talk about the conditions as being probabilities, and a Bor(I)-nameassigns membership in a probabilistic sense. Because of the ready intuition this example can provide, probabilisticlanguage is sometimes used with other forcing posets.

24 CHAPTER 10. FORCING (MATHEMATICS)

10.3 Countable transitive models and generic filters

The key step in forcing is, given a ZFC universe V, to find appropriate G not in V. The resulting class of all interpre-tations of P-names will turn out to be a model of ZFC, properly extending the original V (since G∉V).Instead of working with V, one considers a countable transitive model M with (P,≤,1) ∈M. By model, we mean amodel of set theory, either of all of ZFC, or a model of a large but finite subset of the ZFC axioms, or some variantthereof. Transitivity means that if x ∈ y ∈M, then x ∈M. TheMostowski collapsing theorem says this can be assumedif the membership relation is well-founded. The effect of transitivity is that membership and other elementary notionscan be handled intuitively. Countability of the model relies on the Löwenheim–Skolem theorem.Since M is a set, there are sets not in M – this follows from Russell’s paradox. The appropriate set G to pick, andadjoin toM, is a generic filter on P. The filter condition means that G⊆P and

• 1 ∈ G ;• if p ≥ q ∈ G, then p ∈ G ;• if p,q ∈ G, then ∃r ∈ G, r ≤ p and r ≤ q ;

For G to be generic means

• if D ∈M is a dense subset of P (that is, p ∈ P implies ∃q ∈ D, q ≤ p) then G∩D ≠ 0 .

The existence of a generic filter G follows from the Rasiowa–Sikorski lemma. In fact, slightly more is true: given acondition p ∈ P, one can find a generic filter G such that p ∈ G. Due to the splitting condition, if G is filter, then P\Gis dense. If G is inM then P\G is inM becauseM is model of set theory. By this reason, a generic filter is never inM.

10.4 Forcing

Given a generic filterG⊆P, one proceeds as follows. The subclass ofP-names inM is denotedM(P). LetM[G]={val(u,G):u∈M(P)}.To reduce the study of the set theory ofM[G] to that ofM, one works with the forcing language, which is built uplike ordinary first-order logic, with membership as binary relation and all the names as constants.Define p ⊩M,P φ(u1,…,un) (read "p forces φ in model M with poset P”) where p is a condition, φ is a formulain the forcing language, and the ui are names, to mean that if G is a generic filter containing p, then M[G] ⊨φ(val(u1,G),…,val(un,G)). The special case 1 ⊩M,P φ is often written P ⊩M,P φ or ⊩M,P φ. Such statementsare true inM[G] no matter what G is.What is important is that this “external” definition of the forcing relation p ⊩M,P φ is equivalent to an “internal”definition, defined by transfinite induction over the names on instances of u ∈ v and u = v, and then by ordinaryinduction over the complexity of formulas. This has the effect that all the properties ofM[G] are really properties ofM, and the verification of ZFC inM[G] becomes straightforward. This is usually summarized as three key properties:

• Truth: M[G] ⊨ φ(val(u1,G),…,val(un,G)) if and only if it is forced by G, that is, for some condition p ∈ G, p⊩M,P φ(u1,…,un).

• Definability: The statement "p ⊩M,P φ(u1,…,un)" is definable inM.

• Coherence: If p ⊩M,P φ(u1,…,un) and q ≤ p, then q ⊩M,P φ(u1,…,un).

We define the forcing relation in V by induction on complexity, in which we simultaneously define forcing of atomicformulas by ∈-induction and then we define it by induction on formula complexity.1. p ⊩P a ∈ b if (∀q ≤ p)(∃r ≤ q)(∃s, c)((s, c) ∈ b ∧ r ≤ s ∧ r ⊩P a = c) .2. p ⊩P a = b if (∀q ≤ p)(∀c ∈ V P )(q ⊩P c ∈ a ⇔ q ⊩P c ∈ b) .3. p ⊩P ¬f if ¬(∃q ≤ p)q ⊩P f .4. p ⊩P (f ∧ g) if (p ⊩P f ∧ p ⊩P g) .

10.5. CONSISTENCY 25

5. p ⊩P (∀x)f if (∀x ∈ V P )p ⊩P f .In 1–5 p is an arbitrary condition. In 1 and 2 a and a are arbitrary names and in 3–5 f and g are arbitrary formulaswhere all free occurrences of variables referring names. This definition is syntax transform of formulas. This meansthat for any given formula f(x1, . . . , xn) the formula p ⊨P f(x1, . . . , xn)with free variables p, P, x1, . . . , xn is welldefined. In fact this syntax transform has following properties: any equivalence given by 1-5 is theorem (single theo-rem per formula) and for any formula f(x1, . . . , xn) following formula (∀p, P, x1, . . . , xn)(p ⊩P f(x1, . . . , xn) ⇒(po(P ) ∧ p ∈ dom(P ) ∧ x1, . . . , xn ∈ V P ) is theorem where po(P ) means that P is partial order with splittingcondition. The bit of definition is existence of syntax transform with these properties.This definition provides the possibility of working in V without any countable transitive modelM . The followingstatement gives announced definability:(∀M,P, x1, . . . , xn)(ctm(M)∧po(P )∧P ∈M∧p ∈ dom(P )∧x1, . . . , xn ∈MP ⇒ (p ⊩M,P f(x1, . . . , xn) ⇔M |= p ⊩P f(x1, . . . , xn)))

where ctm(M) means thatM is countable transitive model satisfying some finite part of ZF axioms depending onformula f .(Where no confusion is possible we simply write ⊩ .)

10.5 Consistency

The above can be summarized by saying the fundamental consistency result is that given a forcing poset P, we mayassume that there exists a generic filter G, not in the universe V, such that V[G] is again a set theoretic universe,modelling ZFC. Furthermore, all truths in V[G] can be reduced to truths in V regarding the forcing relation.Both styles, adjoining G to a countable transitive model M or to the whole universe V, are commonly used. Lesscommonly seen is the approach using the “internal” definition of forcing, and no mention of set or class models ismade. This was Cohen’s original method, and in one elaboration, it becomes the method of Boolean-valued analysis.

10.6 Cohen forcing

The simplest nontrivial forcing poset is ( Fin(ω,2), ⊇, 0 ), the finite partial functions from ω to 2={0,1} under reverseinclusion. That is, a condition p is essentially two disjoint finite subsets p−1[1] and p−1[0] of ω, to be thought of asthe “yes” and “no” parts of p, with no information provided on values outside the domain of p. q is stronger than pmeans that q ⊇ p, in other words, the “yes” and “no” parts of q are supersets of the “yes” and “no” parts of p, and inthat sense, provide more information.Let G be a generic filter for this poset. If p and q are both in G, then p∪q is a condition, because G is a filter. Thismeans that g=⋃G is a well-defined partial function from ω to 2, because any two conditions in G agree on theircommon domain.g is in fact a total function. Given n ∈ ω, let Dn={ p : p(n) is defined }, then Dn is dense. (Given any p, if n is not inp’s domain, adjoin a value for n, the result is in Dn.) A condition p ∈ G∩Dn has n in its domain, and since p ⊆ g, g(n)is defined.Let X=g−1[1], the set of all “yes” members of the generic conditions. It is possible to give a name for X directly.Let X = { ( nˇ, p ) : p(n)=1 }, then val( X, G ) = X. Now suppose A⊆ω in V. We claim that X≠A. Let DA = { p :∃n, n∈dom(p) and p(n)=1 if and only if n∉A }. DA is dense. (Given any p, if n is not in p’s domain, adjoin a valuefor n contrary to the status of "n∈A".) Then any p∈G∩DA witnesses X≠A. To summarize, X is a new subset of ω,necessarily infinite.Replacing ω with ω×ω2, that is, consider instead finite partial functions whose inputs are of the form (n,α), with n<ωand α<ω2, and whose outputs are 0 or 1, one gets ω2 new subsets of ω. They are all distinct, by a density argument:given α<β<ω2, let Dα,ᵦ={p:∃n, p(n,α)≠p(n,β)}, then each Dα,ᵦ is dense, and a generic condition in it proves that theαth new set disagrees somewhere with the βth new set.This is not yet the falsification of the continuum hypothesis. One must prove that no new maps have been introducedwhich map ω onto ω1 or ω1 onto ω2. For example, if one considers instead Fin(ω,ω1), finite partial functions fromω to ω1, the first uncountable ordinal, one gets in V[G] a bijection from ω to ω1. In other words, ω1 has collapsed,and in the forcing extension, is a countable ordinal.

26 CHAPTER 10. FORCING (MATHEMATICS)

The last step in showing the independence of the continuum hypothesis, then, is to show that Cohen forcing does notcollapse cardinals. For this, a sufficient combinatorial property is that all of the antichains of this poset are countable.

10.7 The countable chain condition

Main article: Countable chain condition

An antichain A of P is a subset such that if p and q are in A, then p and q are incompatible (written p ⊥ q), meaningthere is no r in P such that r ≤ p and r ≤ q. In the Borel sets example, incompatibility means p∩q has measure zero.In the finite partial functions example, incompatibility means that p∪q is not a function, in other words, p and q assigndifferent values to some domain input.P satisfies the countable chain condition (c.c.c.) if every antichain in P is countable. (The name, which is obviouslyinappropriate, is a holdover from older terminology. Some mathematicians write “c.a.c.” for “countable antichaincondition”.)It is easy to see that Bor(I) satisfies the c.c.c., because the measures add up to at most 1. Fin(E,2) is also c.c.c., butthe proof is more difficult.Given an uncountable subfamilyW ⊆ Fin(E,2), shrinkW to an uncountable subfamilyW0 of sets of size n, for somen<ω. If p(e1)=b1 for uncountably many p ∈W0, shrink to this uncountable subfamilyW1, and repeat, getting a finiteset { (e1,b1), …, (ek,bk) }, and an uncountable family Wk of incompatible conditions of size n−k such that every eis in at most countably many dom(p) for p ∈Wk. Now pick an arbitrary p ∈Wk, and pick fromWk any q that is notone of the countably many members that have a domain member in common with p. Then p ∪ { (e1,b1), …, (ek,bk)} and q ∪ { (e1,b1), …, (ek,bk) } are compatible, so W is not an antichain. In other words, Fin(E,2) antichains arecountable.The importance of antichains in forcing is that for most purposes, dense sets and maximal antichains are equivalent.A maximal antichain A is one that cannot be extended and still be an antichain. This means every element of p ∈ P iscompatible with some member of A. Their existence follows from Zorn’s lemma. Given a maximal antichain A, letD = { p : p≤q, some q∈A }. D is dense, and G∩D≠0 if and only if G∩A≠0. Conversely, given a dense set D, Zorn’slemma shows there exists a maximal antichain A⊆D, and then G∩D≠0 if and only if G∩A≠0.Assume P is c.c.c. Given x,y ∈ V, with f:x→y in V[G], one can approximate f inside V as follows. Let u be a namefor f (by the definition of V[G]) and let p be a condition which forces u to be a function from x to y. Define a functionF whose domain is x by F(a) = { b : ∃ q ≤ p, q forces u(aˇ) = bˇ }. By definability of forcing, this definition makessense within V. By coherence of forcing, different b’s come from incompatible p’s. By c.c.c., F(a) is countable.In summary, f is unknown in V, since it depends on G, but it is not wildly unknown for a c.c.c. forcing. One canidentify a countable set of guesses for what the value of f is at any input, independent of G.This has the following very important consequence. If in V[G], f:α→β is a surjection from one infinite ordinal toanother, then there is a surjection g:ω×α→β in V and consequently a surjection h:α→β in V. In particular, cardinalscannot collapse. The conclusion is that 2ℵ₀ ≥ ℵ2 in V[G].

10.8 Easton forcing

The exact value of the continuum in the above Cohen model, and variants like Fin(ω × κ , 2) for cardinals κ ingeneral, was worked out by Robert M. Solovay, who also worked out how to violate GCH (the generalized continuumhypothesis), for regular cardinals only, a finite number of times. For example, in the above Cohen model, if CH holdsin V, then 2ℵ₀ = ℵ2 holds in V[G].W. B. Easton worked out the infinite and proper class version of violating the GCH for regular cardinals, basicallyshowing the known restrictions (monotonicity, Cantor’s theorem, and König’s theorem) were the only ZFC provablerestrictions. See Easton’s theorem.Easton’s work was notable in that it involved forcing with a proper class of conditions. In general, the method offorcing with a proper class of conditions will fail to give a model of ZFC. For example, Fin ( ω × On , 2 ), where“On” is the proper class of all ordinals, will make the continuum a proper class. Fin ( ω, On ) will introduce acountable enumeration of the ordinals. In both cases, the resulting V[G] is visibly not a model of ZFC.

10.9. RANDOM REALS 27

At one time, it was thought that more sophisticated forcing would also allow arbitrary variation in the powers ofsingular cardinals. But this has turned out to be a difficult, subtle and even surprising problem, with several morerestrictions provable in ZFC, and with the forcing models depending on the consistency of various large cardinalproperties. Many open problems remain.

10.9 Random reals

Main article: random algebra

In the Borel sets ( Bor(I), ⊆, I ) example, the generic filter converges to a real number r, called a random real. Aname for the decimal expansion of r (in the sense of the canonical set of decimal intervals that converge to r) can begiven by letting r = { ( Eˇ, E ) : E = [ k⋅10−n, (k + 1)⋅10−n ], 0 ≤ k < 10n }. This is, in some sense, just a subname ofG.To recover G from r, one takes those Borel subsets of I that “contain” r. Since the forcing poset is in V, but r is not inV, this containment is actually impossible. But there is a natural sense in which the interval [0.5, 0.6] in V “contains”a random real whose decimal expansion begins 0.5. This is formalized by the notion of “Borel code”.Every Borel set can, nonuniquely, be built up, starting from intervals with rational endpoints and applying the opera-tions of complement and countable unions, a countable number of times. The record of such a construction is calleda Borel code. Given a Borel set B in V, one recovers a Borel code, and then applies the same construction sequencein V[G], getting a Borel set B*. One can prove that one gets the same set independent of the construction of B, andthat basic properties are preserved. For example, if B⊆C, then B*⊆C*. If B has measure zero, then B* has measurezero.So given r, a random real, one can show thatG = { B (inV) : r∈B* (inV[G]) }. Because of the mutual interdefinabilitybetween r and G, one generally writes V[r] for V[G].A different interpretation of reals in V[G] was provided by Dana Scott. Rational numbers in V[G] have names thatcorrespond to countably many distinct rational values assigned to a maximal antichain of Borel sets, in other words,a certain rational-valued function on I = [0,1]. Real numbers in V[G] then correspond to Dedekind cuts of suchfunctions, that is, measurable functions.

10.10 Boolean-valued models

Main article: Boolean-valued model

Perhaps more clearly, the method can be explained in terms of Boolean-valued models. In these, any statement isassigned a truth value from some complete atomless Boolean algebra, rather than just a true/false value. Then anultrafilter is picked in this Boolean algebra, which assigns values true/false to statements of our theory. The point isthat the resulting theory has a model which contains this ultrafilter, which can be understood as a new model obtainedby extending the old one with this ultrafilter. By picking a Boolean-valued model in an appropriate way, we can get amodel that has the desired property. In it, only statements which must be true (are “forced” to be true) will be true,in a sense (since it has this extension/minimality property).

10.11 Meta-mathematical explanation

In forcing we usually seek to show some sentence is consistent with ZFC (or optionally some extension of ZFC). Oneway to interpret the argument is that we assume ZFC is consistent and use it to prove ZFC combined with our newsentence is also consistent.Each “condition” is a finite piece of information – the idea is that only finite pieces are relevant for consistency, sinceby the compactness theorem a theory is satisfiable if and only if every finite subset of its axioms is satisfiable. Then,we can pick an infinite set of consistent conditions to extend our model. Thus, assuming consistency of set theory,we prove consistency of the theory extended with this infinite set.

28 CHAPTER 10. FORCING (MATHEMATICS)

10.12 Logical explanation

By Gödel’s incompleteness theorem one cannot prove the consistency of any sufficiently strong formal theory, such asZFC, using only the axioms of the theory itself, unless the theory itself is inconsistent. Consequently mathematiciansdo not attempt to prove the consistency of ZFC using only the axioms of ZFC, or to prove ZFC+H is consistent forany hypothesis H using only ZFC+H. For this reason the aim of a consistency proof is to prove the consistency ofZFC + H relative to consistency of ZFC. Such problems are known as problems of relative consistency. In fact oneproves(*) ZFC ⊢ Con(ZFC) → Con(ZFC +H).

We will give the general schema of relative consistency proofs. Because any proof is finite it uses finite number ofaxioms.

ZFC + ¬Con(ZFC +H) ⊢ ∃T (Fin(T ) ∧ T ⊂ ZFC ∧ (T ⊢ ¬H)).

For any given proof ZFC can verify validity of this proof. This is provable by induction by the length of the proof.

ZFC ⊢ ∀T ((T ⊢ ¬H) → (ZFC ⊢ (T ⊢ ¬H))).

Now we obtain

ZFC + ¬Con(ZFC +H) ⊢ ∃T (Fin(T ) ∧ T ⊂ ZFC ∧ (ZFC ⊢ (T ⊢ ¬H))).

If we prove the following(**) ZFC ⊢ ∀T (Fin(T ) ∧ T ⊂ ZFC → (ZFC ⊢ Con(T +H)))

we can conclude that

ZFC + ¬Con(ZFC +H) ⊢ ∃T (Fin(T ) ∧ T ⊂ ZFC ∧ (ZFC ⊢ (T ⊢ ¬H)) ∧ (ZFC ⊢ Con(T +H)))

which is equivalent to

ZFC + ¬Con(ZFC +H) ⊢ ¬Con(ZFC)

which gives (*). The core of the relative consistency proof is proving (**). One has to construct a ZFC proof ofCon(T + H) for any given finite set T of ZFC axioms (by ZFC instruments of course). (No universal proof of Con(T+ H) of course.)In ZFC it is provable that for any condition p the set of formulas (evaluated by names) forced by p is deductivelyclosed. Also, for any ZFC axiom, ZFC proves that this axiom is forced by 1. Then it suffices to prove that there is atleast one condition which forces H.In the case of Boolean valued forcing, the procedure is similar – one has to prove that the Boolean value of H is not0.Another approach uses the reflection theorem. For any given finite set of ZFC axioms there is ZFC proof that thisset of axioms has a countable transitive model. For any given finite set T of ZFC axioms there is finite set T' of ZFCaxioms such that ZFC proves that if a countable transitive model M satisfies T' then M[G] satisfies T. One has toprove that there is finite set T” of ZFC axioms such that if a countable transitive model M satisfies T” then M[G]satisfies the hypothesis H. Then, for any given finite set T of ZFC axioms, ZFC proves Con(T + H).Sometimes in (**) some stronger theory S than ZFC is used for proving Con(T + H). Then we have proof of consis-tency of ZFC + H relative to the consistency of S. Note that ZFC ⊢ Con(ZFC) ↔ Con(ZFL) , where ZFL is ZF+ V = L (axiom of constructibility).

10.13. SEE ALSO 29

10.13 See also• List of forcing notions

• Nice name

10.14 References• Bell, J. L. (1985) Boolean-ValuedModels and Independence Proofs in Set Theory, Oxford. ISBN 0-19-853241-5

• Cohen, P. J. (1966). Set theory and the continuum hypothesis. Addison–Wesley. ISBN 0-8053-2327-9.

• Grishin, V.N. (2001), “Forcing method”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. North-Holland. ISBN 0-444-85401-0.

10.15 External links• Nik Weaver’s book Forcing for Mathematicians was written for mathematicians who want to learn the basicmachinery of forcing. No background in logic is assumed, beyond the facility with formal syntax which shouldbe second nature to any well-trained mathematician.

• Tim Chow’s article A Beginner’s Guide to Forcing is a good introduction to the concepts of forcing that avoidsa lot of technical detail. This paper grew out of Chow’s newsgroup article Forcing for dummies. In addition toimproved exposition, the Beginner’s Guide includes a section on Boolean Valued Models.

• See also Kenny Easwaran’s article A Cheerful Introduction to Forcing and the Continuum Hypothesis, whichis also aimed at the beginner but includes more technical details than Chow’s article.

• The Independence of the Continuum Hypothesis Paul J. Cohen, Proceedings of the National Academy ofSciences of the United States of America, Vol. 50, No. 6. (Dec. 15, 1963), pp. 1143–1148.

• The Independence of the Continuum Hypothesis, II Paul J. Cohen Proceedings of the National Academy ofSciences of the United States of America, Vol. 51, No. 1. (Jan. 15, 1964), pp. 105–110.

• Paul Cohen gave a historical lecture The Discovery of Forcing (Rocky Mountain J. Math. Volume 32, Number4 (2002), 1071–1100) about how he developed his independence proof. The linked page has a download linkfor an open access PDF but your browser must send a referer header from the linked page to retrieve it.

• Weisstein, Eric W., “Forcing”, MathWorld.

Chapter 11

Generic filter

In the mathematical field of set theory, a generic filter is a kind of object used in the theory of forcing, a tech-nique used for many purposes, but especially to establish the independence of certain propositions from certainformal theories, such as ZFC. For example, Paul Cohen used the method to establish that ZFC, if consistent, cannotprove the continuum hypothesis, which states that there are exactly aleph-one real numbers. In the contemporaryre-interpretation of Cohen’s proof, it proceeds by constructing a generic filter that codes more than ℵ1 reals, withoutchanging the value of ℵ1 .Formally, let P be a poset (partially ordered set), and let F be a filter on P; that is, F is a subset of P such that:

1. F is nonempty

2. If p,q∈P and p≤q and p is an element of F, then q is an element of F (F is closed upward)

3. If p and q are elements of F, then there is an element r of F such that r≤p and r≤q (any two elements of F arecompatible)

Now if D is a collection of dense open subsets of P, in the topology whose basic open sets are all sets of the form{q|q≤p} for particular p in P, then F is said to be D-generic if F meets all sets in D; that is,

F ∩ E ̸= ∅, for all E ∈ D

Similarly, ifM is a transitive model of ZFC (or some sufficient fragment thereof), with P an element ofM, then F issaid to beM-generic, or sometimes generic overM, if F meets all dense open subsets of P that are elements of M.

11.1 References• K. Ciesielski, Set Theory for the Working Mathematician, London Mathematical Society

30

Chapter 12

Iterated forcing

In mathematics, iterated forcing is a method for constructing models of set theory by repeating Cohen’s forcingmethod a transfinite number of times. Iterated forcing was introduced by Solovay and Tennenbaum (1971) in theirconstruction of a model of set theory with no Suslin tree. They also showed that iterated forcing can construct modelswhere Martin’s axiom holds and the continuum is any given regular cardinal.In iterated forcing, one has a transfinite sequence Pα of forcing notions indexed by some ordinals α, which give afamily of Boolean-valued models VPα. If α+1 is a successor ordinal then Pα₊₁ is often constructed from Pα usinga forcing notion in VPα, while if α is a limit ordinal then Pα is often constructed as some sort of limit (such as thedirect limit) of the Pᵦ for β<α.A key consideration is that, typically, it is necessary that ω1 is not collapsed. This is often accomplished by the useof a preservation theorem such as:+ Finite support iteration of c.c.c. forcings (see countable chain condition) are c.c.c. and thus preserve ω1 .+ Countable support iterations of proper forcings are proper (see Fundamental Theorem of Proper Forcing) and thuspreserve ω1 .+ Revised countable support iterations of semi-proper forcings are semi-proper and thus preserve ω1 .Some non-semi-proper forcings, such as Namba forcing, can be iterated with appropriate cardinal collapses whilepreserving ω1 using methods developed by Saharon Shelah[1][2][3]

12.1 References

[1] Shelah, S., Proper and Improper Forcing, Springer 1992

[2] Schlindwein, Chaz, Shelah’s work on non-semiproper iterations I, Archive for Mathematical Logic (47) 2008 pp. 579 --606

[3] Schlindwein, Chaz, Shelah’s work on non-semiproper iterations II, Journal of Symbolic Logic (66) 2001, pp. 1865 -- 1883

• Jech, Thomas (2003), Set Theory: Millennium Edition, Springer Monographs in Mathematics, Berlin, NewYork: Springer-Verlag, ISBN 978-3-540-44085-7

• Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, Elsevier, ISBN 978-0-444-86839-8

• Shelah, Saharon (1998) [1982], Proper and improper forcing, Perspectives in Mathematical Logic (2 ed.),Berlin: Springer-Verlag, ISBN 3-540-51700-6, MR 1623206

• Solovay, R. M.; Tennenbaum, S. (1971). “Iterated Cohen extensions and Souslin’s problem”. Ann. Of Math.(2) (Annals of Mathematics) 94 (2): 201–245. doi:10.2307/1970860. JSTOR 1970860.

31

32 CHAPTER 12. ITERATED FORCING

12.2 External links• Eisworth, Todd; Moore, Justin Tatch (2009), Milovich, David, ed., ITERATED FORCING AND THE CONTIN-

UUM HYPOTHESIS (PDF), Appalachian Set Theory Workshop lecture notes

Chapter 13

Laver property

In mathematical set theory, the Laver property holds between two models if they are not “too dissimilar”, in thefollowing sense.ForM and N transitive models of set theory, N is said to have the Laver property overM if and only if for everyfunction g ∈ M mapping ω to ω \ {0} such that g diverges to infinity, and every function f ∈ N mapping ω to ωand every function h ∈M which bounds f , there is a tree T ∈M such that each branch of T is bounded by h andfor every n the nth level of T has cardinality at most g(n) and f is a branch of T .[1]

A forcing notion is said to have the Laver property if and only if the forcing extension has the Laver property overthe ground model. Examples include Laver forcing.The concept is named after Richard Laver.Shelah proved that when proper forcings with the Laver property are iterated using countable supports, the resultingforcing notion will have the Laver property as well.[2][3]

The conjunction of the Laver property and the ωω -bounding property is equivalent to the Sacks property.

13.1 References[1] Shelah, S., Consistently there is no non-trivial ccc forcing notion with the Sacks or Laver property, Combinatorica, vol. 2,

pp. 309 -- 319, (2001)

[2] Shelah, S., Proper and Improper Forcing, Springer (1992)

[3] C. Schlindwein, Understanding preservation theorems: Chapter VI of Proper and Improper Forcing, I. Archive for Math-ematical Logic, vol. 53, 171–202, Springer, 2014

33

Chapter 14

List of forcing notions

In mathematics, forcing is a method of constructing new modelsM[G] of set theory by adding a generic subset G ofa poset P to a model M. The poset P used will determine what statements hold in the new universe (the 'extension');to force a statement of interest thus requires construction of a suitable P. This article lists some of the posets P thathave been used in this construction.

14.1 Notation• P is a poset with order <.• V is the universe of all sets• M is a countable transitive model of set theory• G is a generic subset of P over M.

14.2 Definitions• P satisfies the countable chain condition if every antichain in P is at most countable. This implies that V and

V[G] have the same cardinals (and the same cofinalities).• A subset D of P is called dense if for every p ∈ P there is some q ∈ D with q ≤ p.• A filter on P is a nonempty subset F of P such that if p < q and p ∈ F then q ∈ F, and if p ∈ F and q ∈ F thenthere is some r ∈ F with r ≤ p and r ≤ q.

• A subset G of P is called generic over M if it is a filter that meets every dense subset of P in M.

14.3 Amoeba forcing

Amoeba forcing is forcing with the amoeba order, and adds a measure 1 set of random reals.

14.4 Cohen forcing

In Cohen forcing (named after Paul Cohen) P is the set of functions from a finite subset of ω2 × ω to {0,1} and p <q if p ⊇ q.This poset satisfies the countable chain condition. Forcing with this poset adds ω2 distinct reals to the model; thiswas the poset used by Cohen in his original proof of the independence of the continuum hypothesis.More generally, one can replace ω2 by any cardinal κ so construct a model where the continuum has size at least κ.Here, the only restriction is that κ does not have cofinality ω.

34

14.5. GRIGORIEFF FORCING 35

14.5 Grigorieff forcing

Grigorieff forcing (after Serge Grigorieff) destroys a free ultrafilter on ω

14.6 Hechler forcing

Hechler forcing (after Stephen Herman Hechler) is used to show that Martin’s axiom implies that every family of lessthan c functions from ω to ω is eventually dominated by some such function.P is the set of pairs (s,E) where s is a finite sequence of natural numbers (considered as functions from a finite ordinalto ω) and E is a finite subset of some fixed set G of functions from ω to ω. The element (s, E) is stronger than (t,F)if t is contained in s, F is contained in E, and if k is in the domain of s but not of t then s(k)>h(k) for all h in F.

14.7 Jockusch–Soare forcing

Forcing withΠ01 classes was invented by Robert Soare and Carl Jockusch to prove, among other results, the low basis

theorem. Here P is the set of nonempty Π01 subsets of 2ω (meaning the sets of paths through infinite, computable

subtrees of 2<ω ), ordered by inclusion.

14.8 Iterated forcing

see also iterated forcingIterated forcing with finite supports was introduced by Solovay and Tennenbaum to show the consistency of Suslin’shypothesis. Easton introduced another type of iterated forcing to determine the possible values of the continuumfunction at regular cardinals. Iterated forcing with countable support was investigated by Laver in his proof of theconsistency of Borel’s conjecture, Baumgartner, who introducedAxiomA forcing, and Shelah, who introduced properforcing. Revised countable support iteration was introduced by Shelah to handle semi-proper forcings, such as Prikryforcing, and generalizations, notably including Namba forcing.

14.9 Laver forcing

Laver forcing was used by Laver to show that Borel’s conjecture, which says that all strong measure zero sets arecountable, is consistent with ZFC. (Borel’s conjecture is not consistent with the continuum hypothesis.)

• P is the set of Laver trees, ordered by inclusion.

A Laver tree p is a subset of the finite sequences of natural numbers such that

• p is a tree: p contains any initial sequence of any element of p

• p has a stem: a maximal node s(p) = s ∈ p such that s ≤ t or t ≤ s for all t in p,

• If t ∈ p and s ≤ t then t has an infinite number of immediate successors tn in p for n ∈ ω.

If G is generic for (P,≤), then the real {s(p) : p ∈ G}, called a Laver-real, uniquely determines G.Laver forcing satisfies the Laver property.

14.10 Levy collapsing

Main article: collapsing algebra

36 CHAPTER 14. LIST OF FORCING NOTIONS

These posets will collapse various cardinals, in other words force them to be equal in size to smaller cardinals.

• Collapsing a cardinal to ω: P is the set of all finite sequences of ordinals less than a given cardinal λ. If λ isuncountable then forcing with this poset collapses λ to ω.

• Collapsing a cardinal to another: P is the set of all functions from a subset of κ of cardinality less than κ toλ (for fixed cardinals κ and λ). Forcing with this poset collapses λ down to κ.

• Levy collapsing: If κ is regular and λ is inaccessible, then P is the set of functions p on subsets of λ× κ withdomain of size less than κ and p(α,ξ)<α for every (α,ξ) in the domain of p. This poset collapses all cardinalsless than λ onto κ, but keeps λ as the successor to κ.

Levy collapsing is named for Azriel Levy.

14.11 Magidor forcing

Amongst many forcing notions developed by Magidor, one of the best known is a generalization of Prikry forcingused to change the cofinality of a cardinal to a given smaller regular cardinal.

14.12 Mathias forcing

• An element of P is a pair consisting of a finite set s of natural numbers and an infinite set A of natural numberssuch that every element of s is less than every element of A. The order is defined by

(t, B) is stronger than (s,A) ((t,B) < (s,A)) if s is an initial segment of t, B is a subset ofA, and t is containedin s ∪ A.

Mathias forcing is named for Adrian Richard David Mathias.

14.13 Namba forcing

Namba forcing (after Kanji Namba) is used to change the cofinality of ω2 to ω without collapsing ω1.

• P is the set of all trees T ⊆ ω<ω2 (nonempty downward closed subsets of the set of finite sequences of ordinals

less than ω2) which have the property that any s in T has an extension in T which has ℵ2 immediate successors.P is ordered by inclusion (i.e., subtrees are stronger conditions). The intersection of all trees in the genericfilter defines a countable sequence which is cofinal in ω2.

Namba' forcing is the subset of P such that there is a node below which the ordering is linear and above which eachnode has ℵ2 immediate successors.Magidor and Shelah proved that if CH holds then a generic object of Namba forcing does not exist in the genericextension by Namba', and vice versa.[1][2]

14.14 Prikry forcing

In Prikry forcing (after Karel Prikrý) P is the set of pairs (s,A) where s is a finite subset of a fixed measurable cardinalκ, and A is an element of a fixed normal measure D on κ. A condition (s,A) is stronger than (t, B) if t is an initialsegment of s, A is contained in B, and s is contained in t ∪ B. This forcing notion can be used to change to cofinalityof κ while preserving all cardinals.

14.15. PRODUCT FORCING 37

14.15 Product forcing

Taking a product of forcing conditions is a way of simultaneously forcing all the conditions.

• Finite products: If P and Q are posets, the product poset P× Q has the partial order defined by (p1, q1) ≤ (p2,q2) if p1 ≤ p2 and q1 ≤ q2.

• Infinite products: The product of a set of posets Pi, i ∈ I, each with a largest element 1 is the set of functionsp on I with p(i) ∈ P(i) and such that p(i) = 1 for all but a finite number of i. The order is given by p ≤ q if p(i)≤ q(i) for all i.

• The Easton product (after William Bigelow Easton) of a set of posets Pi, i ∈ I, where I is a set of cardinalsis the set of functions p on I with p(i) ∈ P(i) and such that for every regular cardinal γ the number of elementsα of γ with p(α) ≠ 1 is less than γ.

14.16 Radin forcing

Radin forcing (after Lon Berk Radin), a technically involved generalization of Magidor forcing, adds a closed, un-bounded subset to some regular cardinal λ.If λ is a sufficiently large cardinal, then the forcing keeps λ regular, measurable, supercompact, etc.

14.17 Random forcing

Main article: random algebra

• P is the set of Borel subsets of [0,1] of positive measure, where p is called stronger than q if it is contained inq. The generic set G then encodes a “random real": the unique real xG in all rational intervals [r,s]V[G] suchthat [r,s]V is in G. This real is “random” in the sense that if X is any subset of [0,1]V of measure 1, lying in V,then xG ∈ X.

14.18 Sacks forcing• P is the set of all perfect trees contained in the set of finite {0,1} sequences. (A tree T is a set of finite sequencescontaining all initial segments of its members, and is called perfect if for any element t of T there is a tree scontaining it so that both s0 and s1 are in T.) A tree p is stronger than q if p is contained in q. Forcing withperfect trees was used by Gerald Enoch Sacks to produce a real a with minimal degree of constructibility.

Sacks forcing has the Sacks property.

14.19 Shooting a fast club

For S a stationary subset of ω1 we set P = {⟨σ,C⟩ : σ is a closed sequence from S and C is a closed unboundedsubset of ω1} , ordered by ⟨σ′, C ′⟩ ≤ ⟨σ,C⟩ iff σ′ end-extends σ and C ′ ⊆ C and σ′ ⊆ σ ∪ C . In V [G] , wehave that

∪{σ : (∃C)(⟨σ,C⟩ ∈ G)} is a closed unbounded subset of S almost contained in each club set in V. ℵ1 is

preserved.

14.20 Shooting a club with countable conditions

For S a stationary subset of ω1 we set P equal to the set of closed countable sequences from S. In V [G] , we have that∪G is a closed unbounded subset of S and ℵ1 is preserved, and if CH holds then all cardinals are preserved.

38 CHAPTER 14. LIST OF FORCING NOTIONS

14.21 Shooting a club with finite conditions

For S a stationary subset of ω1 we set P equal to the set of finite sets of pairs of countable ordinals, such that ifp ∈ P and ⟨α, β⟩ ∈ p then α ≤ β and α ∈ S , and whenever ⟨α, β⟩ and ⟨γ, δ⟩ are distinct elements of p then eitherβ < γ or δ < α . P is ordered by reverse inclusion. In V [G] , we have that {α : (∃β)(⟨α, β⟩ ∈

∪G)} is a closed

unbounded subset of S and all cardinals are preserved.

14.22 Silver forcing

Silver forcing (after Jack Howard Silver) satisfies Fusion, the Sacks property, and is minimal with respect to reals(but not minimal).

14.23 References[1] Shelah, S., Proper and Improper Forcing (Claim XI.4.2), Springer, 1998

[2] Schlindwein, C., Shelah’s work on non-semiproper iterations, I, Archive for Mathematical Logic, vol. 47, no. 6, pp. 579-- 606 (2008)

• Kunen, Kenneth (2011), Set theory, Studies in Logic 34, London: College Publications, ISBN 978-1-84890-050-9, Zbl 1262.03001

• Jech, Thomas (2003), Set Theory: Millennium Edition, Springer Monographs in Mathematics, Berlin, NewYork: Springer-Verlag, ISBN 978-3-540-44085-7

• Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, Elsevier, ISBN 978-0-444-86839-8

• Kunen, Kenneth (2011), Set theory, Studies in Logic 34, London: College Publications, ISBN 978-1-84890-050-9, Zbl 1262.03001

14.24 External links• A.Miller (2009), Forcing Tidbits.

Chapter 15

Martin’s maximum

In set theory, Martin’s maximum, introduced by Foreman, Magidor & Shelah (1988), is a generalization of theproper forcing axiom, which is in turn a generalization of Martin’s axiom. It represents the broadest class of forcingsfor which a forcing axiom is consistent.Martin’s maximum (MM) states that if D is a collection of ℵ1 dense subsets of a notion of forcing that preservesstationary subsets of ω1, then there is a D-generic filter. It is a well known fact that forcing with a ccc notion offorcing preserves stationary subsets of ω1, thus MM extends MA( ℵ1 ). If (P,≤) is not a stationary set preservingnotion of forcing, i.e., there is a stationary subset of ω1, which becomes nonstationary when forcing with (P,≤), thenthere is a collection D of ℵ1 dense subsets of (P,≤), such that there is no D-generic filter. This is why MM is calledthe maximal extension of Martin’s axiom.The existence of a supercompact cardinal implies the consistency of Martin’s maximum.[1] The proof uses Shelah'stheories of semiproper forcing and iteration with revised countable supports.MM implies that the value of the continuum is ℵ2

[2] and that the ideal of nonstationary sets on ω1 is ℵ2 -saturated.[3]It further implies stationary reflection, i.e., if S is a stationary subset of some regular cardinal κ≥ω2 and every elementof S has countable cofinality, then there is an ordinal α<κ such that S∩α is stationary in α. In fact, S contains a closedsubset of order type ω1.

15.1 References[1] Jech (2003) p.684

[2] Jech (2003) p.685

[3] Jech (2003) p.687

• Foreman, M.; Magidor, M.; Shelah, Saharon (1988), “Martin’s maximum, saturated ideals, and nonregular ul-trafilters. I.”,Ann. OfMath. (TheAnnals ofMathematics, Vol. 127, No. 1) 127 (1): 1–47, doi:10.2307/1971415,JSTOR 1971415, MR 0924672, Zbl 0645.03028 correction

• Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (Third millennium ed.), Berlin, NewYork: Springer-Verlag, ISBN 978-3-540-44085-7, Zbl 1007.03002

• Moore, Justin Tatch (2011), “Logic and foundations: the proper forcing axiom”, in Bhatia, Rajendra, Proceedingsof the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. II: In-vited lectures (PDF), Hackensack, NJ: World Scientific, pp. 3–29, ISBN 978-981-4324-30-4, Zbl 1258.03075

39

Chapter 16

Nice name

In set theory, a nice name is used in forcing to impose an upper bound on the number of subsets in the generic model.It is used in the context of forcing to prove independence results in set theory such as Easton’s theorem.

16.1 Formal definition

LetM |= ZFC be transitive, (P, <) a forcing notion inM , and suppose G ⊆ P is generic overM . Then for any P-name inM , τ ,η is a nice name for a subset of τ if η is a P -name satisfying the following properties:(1) dom(η) ⊆ dom(τ)

(2) For all P -names σ ∈M , {p ∈ P|⟨σ, p⟩ ∈ η} forms an antichain.(3) (Natural addition): If ⟨σ, p⟩ ∈ η , then there exists q ≥ p in P such that ⟨σ, q⟩ ∈ τ .

16.2 References• Kenneth Kunen (1980) Set theory: an introduction to independence proofs, Volume 102 of Studies in logic andthe foundations of mathematics (Elsevier) ISBN 0-444-85401-0, p.208

40

Chapter 17

Proper forcing axiom

In the mathematical field of set theory, the proper forcing axiom (PFA) is a significant strengthening of Martin’saxiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings.

17.1 Statement

A forcing or partially ordered set P is proper if for all regular uncountable cardinals λ , forcing with P preservesstationary subsets of [λ]ω .The proper forcing axiom asserts that if P is proper and Dα is a dense subset of P for each α<ω1, then there is afilter G ⊆ P such that Dα ∩ G is nonempty for all α<ω1.The class of proper forcings, to which PFA can be applied, is rather large. For example, standard arguments showthat if P is ccc or ω-closed, then P is proper. If P is a countable support iteration of proper forcings, then P is proper.Crucially, all proper forcings preserve ℵ1 .

17.2 Consequences

PFA directly implies its version for ccc forcings, Martin’s axiom. In cardinal arithmetic, PFA implies 2ℵ0 = ℵ2 .PFA implies any two ℵ1 -dense subsets of R are isomorphic,[1] any two Aronszajn trees are club-isomorphic,[2] andevery automorphism of the Boolean algebra P (ω) /fin is trivial.[3] PFA implies that the Singular Cardinals Hypothesisholds. An especially notable consequence proved by John R. Steel is that the axiom of determinacy holds in L(R),the smallest inner model containing the real numbers. Another consequence is the failure of square principles andhence existence of inner models with many Woodin cardinals.

17.3 Consistency strength

If there is a supercompact cardinal, then there is a model of set theory in which PFA holds. The proof uses the factthat proper forcings are preserved under countable support iteration, and the fact that if κ is supercompact, then thereexists a Laver function for κ .It is not yet known how much large cardinal strength comes from PFA.

17.4 Other forcing axioms

The bounded proper forcing axiom (BPFA) is a weaker variant of PFA which instead of arbitrary dense subsetsapplies only to maximal antichains of size ω1. Martin’s maximum is the strongest possible version of a forcingaxiom.Forcing axioms are viable candidates for extending the axioms of set theory as an alternative to large cardinal axioms.

41

42 CHAPTER 17. PROPER FORCING AXIOM

17.5 The Fundamental Theorem of Proper Forcing

The Fundamental Theorem of Proper Forcing, due to Shelah, states that any countable support iteration of properforcings is itself proper. This follows from the Proper Iteration Lemma, which states that whenever ⟨Pα : α ≤ κ⟩ isa countable support forcing iteration based on ⟨Qα : α < κ⟩ and N is a countable elementary substructure of Hλ

for a sufficiently large regular cardinal λ , and Pκ ∈ N and α ∈ κ ∩ N and p is (N,Pα) -generic and p forces "q ∈ Pκ/GPα ∩N [GPα ] ,” then there exists r ∈ Pκ such that r is N -generic and the restriction of r to Pα equals pand p forces the restriction of r to [α, κ) to be stronger or equal to q .This version of the Proper Iteration Lemma, in which the name q is not assumed to be inN , is due to Schlindwein.[4]

The Proper Iteration Lemma is proved by a fairly straightforward induction on κ , and the Fundamental Theorem ofProper Forcing follows by taking α = 0 .

17.6 See also• Stevo Todorčević

17.7 References[1] Moore (2011)

[2] Abraham, U., and Shelah, S., Isomorphism types of Aronszajn trees (1985) Israel Journal of Mathematics (50) 75 -- 113

[3] Moore (2011)

[4] Schlindwein, C., “Consistency of Suslin’s hypothesis, a non-special Aronszajn tree, and GCH”, (1994), Journal of SymbolicLogic (59) pp. 1 -- 29

• Jech, Thomas (2002). Set theory (Third millennium (revised and expanded) ed.). Springer. doi:10.1007/3-540-44761-X. ISBN 3-540-44085-2. Zbl 1007.03002.

• Kunen, Kenneth (2011). Set theory. Studies in Logic 34. London: College Publications. ISBN 978-1-84890-050-9. Zbl 1262.03001.

• Moore, Justin Tatch (2011). “Logic and foundations: the proper forcing axiom”. In Bhatia, Rajendra. Proceedingsof the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol.II: Invited lectures (PDF). Hackensack, NJ: World Scientific. pp. 3–29. ISBN 978-981-4324-30-4. Zbl1258.03075.

• Steel, JohnR. (2005). “PFA impliesAD^L(R)". Journal of Symbolic Logic 70 (4): 1255–1296. doi:10.2178/jsl/1129642125.

Chapter 18

Ramified forcing

In the mathematical discipline of set theory, ramified forcing is the original form of forcing introduced by Cohen(1963) to prove the independence of the continuum hypothesis from Zermelo–Fraenkel set theory. Ramified forcingstarts with a model M of set theory in which the axiom of constructibility, V = L, holds, and then builds up a largermodel M[G] of Zermelo–Fraenkel set theory by adding a generic subset G of a partially ordered set to M, imitatingKurt Gödel's constructible hierarchy.Dana Scott and Robert Solovay realized that the use of constructible sets was an unnecessary complication, and couldbe replaced by a simpler construction similar to John von Neumann's construction of the universe as a union of setsR(α) for ordinals α. Their simplification was originally called “unramified forcing” (Shoenfield 1971), but is nowusually just called “forcing”. As a result, ramified forcing is only rarely used.

18.1 References• Cohen, P. J. (1966), Set Theory and the Continuum Hypothesis, Menlo Park, CA: W. A. Benjamin.

• Cohen, Paul J. (1963), “The Independence of the ContinuumHypothesis”, Proceedings of the National Academyof Sciences of the United States of America 50 (6): 1143–1148, doi:10.1073/pnas.50.6.1143, ISSN 0027-8424,JSTOR 71858, PMC 221287, PMID 16578557.

• Shoenfield, J. R. (1971), “Unramified forcing”, Axiomatic Set Theory, Proc. Sympos. Pure Math., XIII, PartI, Providence, R.I.: Amer. Math. Soc., pp. 357–381, MR 0280359.

43

Chapter 19

Random algebra

In set theory, the random algebra or random real algebra is the Boolean algebra of Borel sets of the unit intervalmodulo the ideal of measure zero sets. It is used in random forcing to add random reals to a model of set theory.The random algebra was studied by John von Neumann in 1935 (in work later published as Neumann (1998, p. 253))who showed that it is not isomorphic to the Cantor algebra of Borel sets modulo meager sets. Random forcing wasintroduced by Solovay (1970).

19.1 References• Bartoszynski, Tomek (2010), “Invariants of measure and category”, Handbook of set theory 2, Springer, pp.491–555, MR 2768686

• Bukowský, Lev (1977), “Random forcing”, Set theory and hierarchy theory, V (Proc. Third Conf., Bierutowice,1976), Lecture Notes in Math. 619, Berlin: Springer, pp. 101–117, MR 0485358

• Solovay, Robert M. (1970), “Amodel of set-theory in which every set of reals is Lebesgue measurable”, Annalsof Mathematics. Second Series 92: 1–56, ISSN 0003-486X, JSTOR 1970696, MR 0265151

• Neumann, John von (1998) [1960], Continuous geometry, Princeton Landmarks in Mathematics, PrincetonUniversity Press, ISBN 978-0-691-05893-1, MR 0120174

44

Chapter 20

Rasiowa–Sikorski lemma

In axiomatic set theory, the Rasiowa–Sikorski lemma (named after Helena Rasiowa and Roman Sikorski) is one ofthe most fundamental facts used in the technique of forcing. In the area of forcing, a subset D of a forcing notion (P,≤) is called dense in P if for any p ∈ P there is d ∈ D with d ≤ p. A filter F in P is called D-generic if

F ∩ E ≠ ∅ for all E ∈ D.

Now we can state the Rasiowa–Sikorski lemma:

Let (P, ≤) be a poset and p ∈ P. If D is a countable family of dense subsets of P then there exists aD-generic filter F in P such that p ∈ F.

20.1 Proof of the Rasiowa–Sikorski lemma

The proof runs as follows: since D is countable, one can enumerate the dense subsets of P as D1, D2, …. Byassumption, there exists p ∈ P. Then by density, there exists p1 ≤ p with p1 ∈ D1. Repeating, one gets … ≤ p2 ≤ p1≤ p with pi ∈ Di. Then G = { q ∈ P: ∃ i, q ≥ pi} is a D-generic filter.The Rasiowa–Sikorski lemma can be viewed as a weaker form of an equivalent to Martin’s axiom. More specifically,it is equivalent to MA( ℵ0 ).

20.2 Examples

• For (P, ≥) = (Func(X, Y), ⊂), the poset of partial functions from X to Y, define Dx = {s ∈ P: x ∈ dom(s)}. IfX is countable, the Rasiowa–Sikorski lemma yields a {Dx: x ∈ X}-generic filter F and thus a function ∪ F: X→ Y.

• If we adhere to the notation used in dealing with D-generic filters, {H ∪ G0: PijPt} forms an H-generic filter.

• If D is uncountable, but of cardinality strictly smaller than 2ℵ0 and the poset has the countable chain condition,we can instead use Martin’s axiom.

20.3 See also

• Generic filter

• Martin’s axiom

45

46 CHAPTER 20. RASIOWA–SIKORSKI LEMMA

20.4 References• Ciesielski, Krzysztof (1997). Set theory for the working mathematician. London Mathematical Society StudentTexts 39. Cambridge: Cambridge University Press. ISBN 0-521-59441-3. Zbl 0938.03067.

• Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Studies in Logic and the Foun-dations of Mathematics 102. North-Holland. ISBN 0-444-85401-0. Zbl 0443.03021.

20.5 External links• Tim Chow’s newsgroup article Forcing for dummies is a good introduction to the concepts and ideas behindforcing; it covers the main ideas, omitting technical details

Chapter 21

Sacks property

In mathematical set theory, the Sacks property holds between two models of Zermelo–Fraenkel set theory if theyare not “too dissimilar” in the following sense.ForM and N transitive models of set theory, N is said to have the Sacks property overM if and only if for everyfunction g ∈ M mapping ω to ω \ {0} such that g diverges to infinity, and every function f ∈ N mapping ω to ωthere is a tree T ∈M such that for every n the nth level of T has cardinality at most g(n) and f is a branch of T .[1]

The Sacks property is used to control the value of certain cardinal invariants in forcing arguments. It is named forGerald Enoch Sacks.A forcing notion is said to have the Sacks property if and only if the forcing extension has the Sacks property overthe ground model. Examples include Sacks forcing and Silver forcing.Shelah proved that when proper forcings with the Sacks property are iterated using countable supports, the resultingforcing notion will have the Sacks property as well.[2][3]

The Sacks property is equivalent to the conjunction of the Laver property and the ωω -bounding property.

21.1 References[1] Shelah, Saharon (2001), “Consistently there is no non trivial ccc forcing notionwith the Sacks or Laver property”, Combinatorica

21 (2): 309–319, doi:10.1007/s004930100027, MR 1832454.

[2] Shelah, Saharon (1998), Proper and improper forcing, Perspectives in Mathematical Logic (2nd ed.), Springer-Verlag,Berlin, doi:10.1007/978-3-662-12831-2, ISBN 3-540-51700-6, MR 1623206.

[3] Schlindwein, Chaz (2014), “Understanding preservation theorems: chapter VI of Proper and improper forcing, I”, Archivefor Mathematical Logic 53 (1-2): 171–202, doi:10.1007/s00153-013-0361-8, MR 3151404

47

Chapter 22

Sunflower (mathematics)

A mathematical sunflower can be pictured as a flower. The kernel of the sunflower is the brown part in the middle, and each set ofthe sunflower is the union of a petal and the kernel.

In mathematics, a sunflower or Δ system is a collection of sets whose pairwise intersection is constant, and calledthe kernel.

48

22.1. FORMAL DEFINITION 49

The Δ lemma, sunflower lemma, and sunflower conjecture give various conditions that imply the existence of alarge sunflower in a given collection of sets.The original term for this concept was "Δ-system”. More recently the term “sunflower”, possibly introduced by Deza& Frankl (1981), has been gradually replacing it.

22.1 Formal definition

Suppose U is a universe set and W is a collection of subsets of U. The collection W is a sunflower (or Δ-system) ifthere is a subset S of U such that for each distinct A and B inW, we have A ∩ B = S. In other words,W is a sunflowerif the pairwise intersection of each set inW is constant.

22.2 Δ lemma

The Δ lemma states that every uncountable collection of finite sets contains an uncountable Δ-system.The Δ lemma is a combinatorial set-theoretic tool used in proofs to impose an upper bound on the size of a collectionof pairwise incompatible elements in a forcing poset. It may for example be used as one of the ingredients in a proofshowing that it is consistent with Zermelo-Fraenkel set theory that the continuum hypothesis does not hold. It wasintroduced by Shanin (1946).

22.3 Δ lemma for ω2

IfW is an ω2 -sized collection of countable subsets of ω2 , and if the continuum hypothesis holds, then there is anω2 -sized Δ-subsystem. Let ⟨Aα : α < ω2⟩ enumerateW . For cf(α) = ω1 , let f(α) = sup(Aα ∩ α) . By Fodor’slemma, fix S stationary in ω2 such that f is constantly equal to β on S . Build S′ ⊆ S of cardinality ω2 such thatwhenever i < j are in S′ then Ai ⊆ j . Using the continuum hypothesis, there are only ω1 -many countable subsetsof β , so by further thinning we may stabilize the kernel.

22.4 Sunflower lemma and conjecture

Erdős &Rado (1960, p. 86) proved the sunflower lemma, stating that if a and b are positive integers then a collectionof b!ab+1 sets of cardinality at most b contains a sunflower with more than a sets. The sunflower conjecture is oneof several variations of the conjecture of (Erdős & Rado 1960, p. 86) that the factor of b! can be replaced by Cb forsome constant C.

22.5 References• Deza,M.; Frankl, P. (1981), “Every large set of equidistant (0,+1,–1)-vectors forms a sunflower”, Combinatorica1 (3): 225–231, doi:10.1007/BF02579328, ISSN 0209-9683, MR 637827

• Erdős, Paul; Rado, R. (1960), “Intersection theorems for systems of sets”, Journal of the London MathematicalSociety, Second Series 35 (1): 85–90, doi:10.1112/jlms/s1-35.1.85, ISSN 0024-6107, MR 0111692

• Jech, Thomas (2003). Set Theory. Springer.

• Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. North-Holland. ISBN 0-444-85401-0.

• Shanin, N. A. (1946), “A theorem from the general theory of sets”, C. R. (Doklady) Acad. Sci. URSS (N.S.)53: 399–400

Chapter 23

Suslin algebra

In mathematics, a Suslin algebra is a Boolean algebra that is complete, atomless, countably distributive, and satisfiesthe countable chain condition. They are named after Mikhail Yakovlevich Suslin.The existence of Suslin algebras is independent of the axioms of ZFC, and is equivalent to the existence of Suslintrees or Suslin lines.

23.1 References• Jech, Thomas (2003). Set theory (third millennium (revised and expanded) ed.). Springer-Verlag. ISBN 3-540-44085-2. OCLC 174929965. Zbl 1007.03002.

50

23.2. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 51

23.2 Text and image sources, contributors, and licenses

23.2.1 Text• Amoeba order Source: https://en.wikipedia.org/wiki/Amoeba_order?oldid=630530253 Contributors: Michael Hardy, R.e.b., Deltahe-

dron and Anonymous: 1• Boolean-valued model Source: https://en.wikipedia.org/wiki/Boolean-valued_model?oldid=680351872 Contributors: Michael Hardy,

TakuyaMurata, Charles Matthews, Tobias Bergemann, Giftlite, Marcos, Ryan Reich, BD2412, R.e.b., Mathbot, Trovatore, SmackBot,Mhss, Ligulembot, Mets501, Zero sharp, CBM, Gregbard, Cydebot, Newbyguesses, Alexey Muranov, Addbot, Lightbot, Citation bot,DrilBot, Kiefer.Wolfowitz and Anonymous: 5

• Cantor algebra Source: https://en.wikipedia.org/wiki/Cantor_algebra?oldid=615945837 Contributors: R.e.b.• Cohen algebra Source: https://en.wikipedia.org/wiki/Cohen_algebra?oldid=683514051 Contributors: Michael Hardy, R.e.b. and David

Eppstein• Collapsing algebra Source: https://en.wikipedia.org/wiki/Collapsing_algebra?oldid=615789032 Contributors: R.e.b., David Eppstein

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Hardy, Silverfish, Charles Matthews, Giftlite, AshtonBenson, Jemiller226, R.e.b., Mathbot, Scythe33, Shanel, Trovatore, Closedmouth,SmackBot, Melchoir, Mhss, Mets501, Zero sharp, Vaughan Pratt, CBM, Cydebot, Headbomb, Noobeditor, Tim Ayeles, Hans Adler,Addbot, Angelobear, Citation bot, VictorPorton, Qm2008q, Citation bot 1, Helpful Pixie Bot, K9re11 and Anonymous: 6

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• Forcing (mathematics) Source: https://en.wikipedia.org/wiki/Forcing_(mathematics)?oldid=669357287Contributors: AxelBoldt, MichaelHardy, Ottawakungfu, Bogdangiusca, Poor Yorick, Charles Matthews, Dysprosia, Quux, Aleph4, Tobias Bergemann, Giftlite, Sam Hoce-var, Splatty, Luqui, Gauge, Peter M Gerdes, Oleg Alexandrov, Ryan Reich, Eyu100, Mikedelsol, R.e.b., Billjefferys, Hornplease, KSmrq,Gaius Cornelius, Trovatore, Baarslag, That Guy, From That Show!, AndrewWTaylor, SmackBot, SaxTeacher, Slaniel, Mhss, Silly rab-bit, Nbarth, Radagast83, Luke Gustafson, Nat2, Beetstra, Mets501, Stotr~enwiki, Jason22~enwiki, JRSpriggs, CRGreathouse, Myasuda,Gregbard, Cydebot, Yrodro, Thijs!bot, Eleuther, Alphachimpbot, S vatev, Ttwo, VolkovBot, Nxavar, YohanN7, Mathwizard44, PaulClapham, ClueBot, Arakunem, MystBot, Good Olfactory, Addbot, שי ,דוד Citation bot, RJGray, VladimirReshetnikov, Ash870284,D'ohBot, MarcelB612, ToematoeAdmn, EmausBot, Set theorist, Fairflow, ChuispastonBot, ClueBot NG, Frietjes, Harsimaja, Ansatz,Nedeljko Stefanovic, ChrisGualtieri, Dtotoo and Anonymous: 47

• Generic filter Source: https://en.wikipedia.org/wiki/Generic_filter?oldid=607158000 Contributors: Michael Hardy, Charles Matthews,Tobias Bergemann, Trovatore, Baarslag, Cydebot, Hans Adler, MystBot, Addbot, Yobot, Chimpionspeak and Anonymous: 1

• Iterated forcing Source: https://en.wikipedia.org/wiki/Iterated_forcing?oldid=635004173 Contributors: R.e.b., Bgwhite, Magioladitis,Yobot, Monkbot and Anonymous: 1

• Laver property Source: https://en.wikipedia.org/wiki/Laver_property?oldid=648614379 Contributors: Michael Hardy, Dodger67, Tku-vho, Brirush and Anonymous: 1

• List of forcing notions Source: https://en.wikipedia.org/wiki/List_of_forcing_notions?oldid=680566228 Contributors: AxelBoldt, TheAnome, Michael Hardy, Aleph4, R.e.b., Cydebot, Kope, Cobi, WereSpielChequers, Pit-trout, ClueBot, Yobot, Omnipaedista, Set theorist,BG19bot, StarryGrandma, Deltahedron and Anonymous: 19

• Martin’smaximum Source: https://en.wikipedia.org/wiki/Martin’{}s_maximum?oldid=621610777Contributors: Michael Hardy, CharlesMatthews, Rjwilmsi, R.e.b., CBM, Cydebot, Kope, Hans Adler, Citation bot, Citation bot 1, DrilBot, Deltahedron and Anonymous: 1

• Nice name Source: https://en.wikipedia.org/wiki/Nice_name?oldid=645469026 Contributors: Michael Hardy, Rich Farmbrough, C S,Bhny, Arthur Rubin, Radagast83, Nagle, Jason22~enwiki, JRSpriggs, CBM,Cydebot, Alphachimpbot, Tomaxer, HansAdler, AnomieBOT,Erik9bot, DARTH SIDIOUS 2, ClueBot NG, Helpful Pixie Bot, Ansatz and Anonymous: 3

52 CHAPTER 23. SUSLIN ALGEBRA

• Proper forcing axiom Source: https://en.wikipedia.org/wiki/Proper_forcing_axiom?oldid=670670192 Contributors: Aleph4, TobiasBergemann, Giftlite, R.e.b., Bubba73, Trovatore, SmackBot, Stotr~enwiki, JRSpriggs, Ntsimp, Alphachimpbot, Gay Cdn, Misc. Math,Popopp, Miaoku, Addbot, DOI bot, Yobot, Citation bot, FrescoBot, Ver-bot, RedBot, Set theorist, BG19bot, Deltahedron, Frogfol andAnonymous: 4

• Ramified forcing Source: https://en.wikipedia.org/wiki/Ramified_forcing?oldid=476916548Contributors: Michael Hardy, Tobias Berge-mann, Rjwilmsi, R.e.b., Trovatore, CBM, Cydebot, David Eppstein, Mild Bill Hiccup, Hans Adler, Citation bot, Citation bot 1 andAnonymous: 1

• Random algebra Source: https://en.wikipedia.org/wiki/Random_algebra?oldid=645415032 Contributors: R.e.b. and Moonraker• Rasiowa–Sikorski lemma Source: https://en.wikipedia.org/wiki/Rasiowa%E2%80%93Sikorski_lemma?oldid=610403906 Contribu-

tors: Michael Hardy, Dominus, CharlesMatthews, Aleph4, Giftlite, Gauge, Algebraist, Trovatore, Baarslag, FF2010, That Guy, FromThatShow!, CapitalR, CBM, Cydebot, Ntsimp, Addbot, Yobot, , Citation bot, Omnipaedista, VladimirReshetnikov, RedBot, EmausBot,ZéroBot, Deltahedron and Anonymous: 6

• Sacks property Source: https://en.wikipedia.org/wiki/Sacks_property?oldid=647943686Contributors: David Eppstein, Rankersbo, Blueras-berry, BattyBot, Mark viking, Sam Sailor, Johanna, MicroPaLeo and Anonymous: 1

• Sunflower (mathematics) Source: https://en.wikipedia.org/wiki/Sunflower_(mathematics)?oldid=672908350Contributors: Michael Hardy,Charles Matthews, David Haslam, R.e.b., J. Finkelstein, Jason22~enwiki, Cydebot, David Eppstein, VolkovBot, Rei-bot, Hans Adler, Ad-dbot, Luckas-bot, Yobot, Citation bot, Trappist the monk, EmausBot, Set theorist, ZéroBot and Anonymous: 4

• Suslin algebra Source: https://en.wikipedia.org/wiki/Suslin_algebra?oldid=616031609 Contributors: R.e.b. and David Eppstein

23.2.2 Images• File:CardContin.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/75/CardContin.svg License: Public domain Contrib-

utors: en:Image:CardContin.png Original artist: en:User:Trovatore, recreated by User:Stannered• File:Edit-clear.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg License: Public domain Contributors: The

Tango! Desktop Project. Original artist:The people from the Tango! project. And according to themeta-data in the file, specifically: “Andreas Nilsson, and Jakub Steiner (althoughminimally).”

• File:Sonnenblume_02_KMJ.jpg Source: https://upload.wikimedia.org/wikipedia/commons/e/ec/Sonnenblume_02_KMJ.jpg License:CC-BY-SA-3.0 Contributors: Original text: “selbst fotografiert” Original artist: KMJ at German Wikipedia

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