The multiverse perspective on determinateness in set theorylogic.harvard.edu/EFI_Hamkins_MultiverseSlides.pdf · The multiverse view Ontology of Forcing Case Studies: CH, V = L Multiverse

The multiverse view Ontology of Forcing Case Studies: CH, V = L Multiverse Axioms Multiverse Mathematics

The multiverse perspective ondeterminateness in set theory

Joel David Hamkins

New York University, Philosophy

The City University of New York, MathematicsCollege of Staten Island of CUNY

The CUNY Graduate Center

Exploring the Frontiers of IncompletenessHarvard University, October 19, 2011

The multiverse perspective, Harvard EFI 2011 Joel David Hamkins, New York


Introduction

Set theory as Ontological FoundationSet theorists commonly take their subject to serve as afoundation for the rest of mathematics, in the sense that otherabstract mathematical objects can be construed fundamentallyas sets, and in this way, they regard the set-theoretical universeas the realm of all mathematics.

On this view, mathematical objects—such as functions, groups,real numbers, ordinals—are sets, and being precise inmathematics amounts to specifying an object in set theory.

A slightly weakened position remains compatible withstructuralism, where the set-theorist holds that sets provideobjects fulfilling the desired structural roles of mathematicalobjects, which therefore can be viewed as sets.



Introduction

The Set-Theoretical Universe

These sets accumulate transfinitely to form the universe of allsets. This cumulative universe is regarded by set theorists asthe domain of all mathematics.

The orthodox view among set theorists thereby exhibits atwo-fold realist or Platonist nature:

First, mathematical objects (can) exist as sets, andSecond, these sets enjoy a real mathematical existence,accumulating to form the universe of all sets.

A principal task of set theory, on this view, is to discover thefundamental truths of this cumulative set-theoretical universe.These truths will include all mathematical truths.



Introduction

Uniqueness of the universe

I would like to emphasize that on this view, the set-theoreticuniverse is held to be unique: it is the universe of all sets.

And interesting set-theoretic questions, such as the ContinuumHypothesis and others, will have definitive final answers in thisuniverse.

The fundamental problem for set-theorists on this view is: howshall we discover these final truths?



Introduction

The Universe View

Let me therefore describe as the universe view, the positionthat:

There is a unique absolute background concept of set,instantiated in the cumulative universe of all sets, in whichset-theoretic assertions have a definite truth value.

Thus, the universe view is one of determinism for set-theoretictruth, and hence also for mathematical truth.



Introduction

Support for the universe view

Adherents of the universe view often point to the increasinglystable body of regularity features flowing from the large cardinalhierarchy, such as the attractive determinacy consequencesand the accompanying uniformization and Lebesguemeasurability properties in the projective hierarchy of sets ofreals.

Because these regularity features are both mathematicallydesirable and highly explanatory, and the large cardinalperspective seems to provide a coherent unifying theory ofthem, this situation is often taken as evidence that we are onthe right track towards the final answers to these set theoreticalquestions.



Introduction

Independence phenomenon as distractionMeanwhile, the pervasive independence phenomenon in settheory, by which we have come to know that numerousset-theoretic questions cannot be settled on the basis of ZFCand other stronger axiomatizations, is viewed as a distraction.

On this view, independence is an irrelevant side discussionabout provability rather than truth. Independence is about theweakness of our theories in finding the truth, rather than aboutthe truth itself, for the independence of a set-theoretic assertionfrom ZFC tells us little about whether it holds or not in theuniverse.

One expects independence, when one has only a weak theory.Thus, we need to strengthen it.



Introduction

An opening thrust against the Universe view

A difficulty for the Universe view. While adherents hold thatthere is a single universe of all sets, nevertheless the mostpowerful set-theoretical tools developed in the past half-centuryare most naturally understood as methods of constructingalternative set-theoretical universes, universes that seemfundamentally set-theoretic.

forcing, ultrapowers, canonical inner models, etc.

Much of set theory over the past half-century has been aboutconstructing as many different models of set theory as possible.These models are often made to exhibit precise, exactingfeatures or to exhibit specific relationships with other models.



Introduction

An imaginary alternative historyImagine that set theory had followed a different history:

Imagine that as set theory developed, theorems wereincreasingly settled in the base theory....that the independence phenomenon was limited toparadoxical-seeming meta-logic statements....that the few true independence results occurring weresettled by missing natural self-evident set principles....that the basic structure of the set-theoretic universebecame increasingly stable and agreed-upon.

Such developments would have constituted evidence for theUniverse view.

But the actual history is not like this...



Introduction

Actual history: an abundance of universesOver the past half-century, set theorists have discovered a vastdiversity of models of set theory, a chaotic jumble ofset-theoretic possibilities.

Whole parts of set theory exhaustively explore thecombinations of statements realized in models of set theory,and study the methods supporting this exploration.

Would you like CH + ¬♦? Do you want 2ℵn = ℵn+2 for all n?Suslin trees? Would you like to separate the bounding numberfrom the dominating number? Do you want an indestructiblestrongly compact cardinal? Would you prefer that it is also theleast measurable cardinal?

We build models to order.The multiverse perspective, Harvard EFI 2011 Joel David Hamkins, New York


Introduction

Category-theoretic natureAs a result, the fundamental object of study in set theory hasbecome: the model of set theory.

We have L, L[0]], L[µ], L[~E ]; we have models V with largecardinals, forcing extensions V [G], ultrapowers M, cut-offuniverses Lδ, Vα, Hκ, universes L(R), HOD, genericultrapowers, boolean ultrapowers, etc. Forcing especially hasled to a staggering variety of models.

Set theory has discovered an entire cosmos of set-theoreticaluniverses, connected by forcing or large cardinal embeddings,like lines in a constellation filling a dark night sky.

Set theory now exhibits a category-theoretic nature.



Introduction

A challenge to the universe view

The challenge is for the universe view to explain this centralphenomenon, the phenomenon of the enormous diversity ofset-theoretic possibilities.

The universe view seems to be fundamentally at odds with theexistence of alternative set theoretic universes. Will they beexplained away as imaginary?

In particular, it does not seem sufficient, when arguing for theuniverse view, to identify a particularly robust or clarifyingtheory, if the competing alternatives still appear acceptablyset-theoretic. It seems that one must still explicitly explain (orexplain away) the pluralistic illusion.



Introduction

The Multiverse View

A competing philosophical position accepts the alternative setconcepts as fully real.

The Multiverse view. The philosophical position holding thatthere are many set-theoretic universes.

The view is that there are numerous distinct concepts of set,not just one absolute concept of set, and each corresponds tothe universe of sets to which it gives rise.



Introduction

Diverse set concepts

The various concepts of set are simply those giving rise to theuniverses we have been constructing.

A key observation

From any given concept of set, we are able to generate manynew concepts of set, relative to it.

From a set concept giving rise to a universe W , we describeother universes LW , HODW , L(R)W , K W , forcing extensionsW [G], W [H], ultrapowers, and so on.

Each such universe amounts to a new concept of set describedin relation to the concept giving rise to W .



Introduction

A philosophical enterprise becomes mathematical

Many of these concepts of set are closely enough related to beanalyzed together from the perspective of a single set concept.

So what might have been a purely philosophicalenterprise—comparing different concepts of set—becomes inpart a mathematical one.

This explains why the emerging area of the philosophy of settheory is such a fascinating mix of extremely technicalmathematical ideas with profoundly philosophical matters.



Introduction

Multiverse view is realism

The multiverse view is a brand of realism. The alternativeset-theoretical universes arise from different concepts of set,each giving rise to a universe of sets fully as real as theUniverse of sets on the Universe view.

The view in part is that our mathematical tools—forcing,etc.—have offered us glimpses into these other mathematicalworlds, providing evidence that they exist.

A Platonist may object at first, but actually, this IS a kind ofPlatonism, namely, Platonism about universes, second-orderrealism. Set theory is mature enough to adopt and analyze thisview mathematically.



Introduction

The MultiverseThe multiverse view places a focus on the connections betweenthe various universes, such as the relations of ground model toforcing extension or model to ultrapower. (And this is oneimportant way it is distinguished from formalism.)

For example, early forcing results were usually stated asrelative consistency results by starting with a model of ϕ andproviding ψ in a forcing extension.

Con(ZFC + ϕ)→ Con(ZFC + ψ)

Contemporary work often states theorems as:If ϕ, then there is a forcing extension with ψ.

Such a policy retains information connecting the models.



Do forcing extensions of the universe exist?

The central dispute is whether there are universes outside V ,taken under the Universe view to consist of all sets. A specialcase captures the essential debate:

Question

Do forcing extensions of the universe exist?

On the Universe view, forcing extensions of V are illusory. Onthe Multiverse view, V is an introduced constant, referring tothe universe currently under consideration.



How forcing worksBefore proceeding, let’s say a bit more about how forcing works.

We start with a model V of set theory and a partial order P in V .

Suppose G ⊆ P is an V -generic filter, meaning that Gcontains members from every dense subset of P in V .

V ⊆ V [G]

The forcing extension V [G] is obtained by closing underelementary set-building operations. Every object in V [G] has aname in V and is constructed directly from its name and G.

Remarkably, the forcing extension V [G] is always a model ofZFC. But it can exhibit different set-theoretic truths in a way thatcan be precisely controlled by the choice of P.



Forcing as projections of multi-valued set theory

Another way to view the forcing extensions is as the classicalprojections of a multi-valued logic approach to set theory.

That is, for any complete Boolean algebra B, we may undertaketo do set theory using B-valued truth. This leads directly to theconcept of B-names as sets, and to the B-valued universe VB.

The projections of VB to classical models via an ultrafilterU ⊆ B are in essence the forcing extensions.

In this way, the set concept of a forcing extension by B is simplya classical projection of the corresponding multi-valued settheory.



Forcing over countable modelsTraditionally, forcing was often used with countable transitiveground models.

One starts with a countable transitive model M of set theory.Since M is countable, it has only countably many dense sets,and so we easily construct an M-generic filter G bydiagonalization. Thus, we construct the forcing extension M[G]very concretely.

Drawbacks:Metamathematical issues with existence of countabletransitive models of ZFC.Limited to forcing over only some models.Pushes much of the analysis into the meta-theory.



Forcing over V

A more sophisticated approach to forcing develops thetechnique within ZFC, so that one can force over any model ofset theory.

We want to make the move from forcing over a model M toforcing over the universe V .

“Removing the training wheels”

There are a variety of ways to accomplish this.



Ontological status of generic filters

Those who take V as the unique universe of all sets object:

“There are no V-generic filters”

Surely we all agree that for nontrivial forcing notions, there areno V -generic filters in V .

But isn’t the objection like saying:

“There is no square root of −1”

Of course,√−1 does not exist in the reals R. One must go to a

field extension, the complex numbers, to find it.

Similarly, one must go to the forcing extension V [G] to find G.



Imaginary objects

Historically,√−1 was viewed with suspicion, deemed

imaginary, but useful.

Eventually, mathematicians realized how to simulate thecomplex numbers a + bi ∈ C inside the real numbers,representing them as pairs (a,b), and thereby gain access tothe complex numbers from a world having only real numbers.

The case of forcing is similar. We don’t have the generic filter Ginside V , but we have a measure of access to the forcingextension V [G] from V .



Naturalist Account of ForcingSuppose that V is a universe of set theory and P is a notion offorcing. Then there is a class model of the theory expressingwhat it means to be the corresponding forcing extension.

Namely, the theory keeps everything true in V , relativized to aclass predicate, and also asserts that there is a V -generic filter.

The language has ∈, constant symbols for every element of V ,a predicate for V , and constant symbol G. The theory asserts:

1 The full elementary diagram of V , relativized to predicate.2 The assertion that V is a transitive proper class.3 G is a V -generic ultrafilter on P.4 ZFC holds, and the (new) universe is V [G].

In practice, this is how forcing arguments proceed“Let G be V-generic. Argue now in V [G].”



Naturalist Account

Another way to describe it is:

For any forcing notion P, the universe V can be embedded

V - V ⊆ V [G]

into a class model V having a V -generic filter G ⊆ P.

The point here is that the entire model V [G] and the embeddingexist as classes in V .

The map V - V is a Boolean ultrapower map.



Analogy between set theory and geometry

The analogy with Geometry

The one true Universe of sets is to The set-theoretic multiverse

As

The one true Geometry is to The spectrum of Euclidean,non-Euclidean geometries




The analogy with GeometryGeometry studied concepts—points, lines, planes—with aseemingly clear, absolute meaning; but those fundamentalconcepts shattered via non-Euclidean geometry into distinctgeometrical concepts, realized in distinct geometrical universes.

The first consistency arguments for non-Euclidean geometrypresented them as simulations within Euclidean geometry (e.g.‘line’ = great circle on sphere).

In time, geometers accepted the alternative geometries morefully, with their own independent existence, and developedintuitions about what it is like to live inside them.

Today, geometers have a deep understanding of thesealternative geometries, and no-one now regards the alternativegeometries as illusory.




Set theory – geometry

Geometers reason about the various geometries:externally, as embedded spaces.internally, by using newly formed intuitions.abstractly, using isometry groups.

Similar modes of reasoning arise with a forcing extension V [G]:

We understand V [G] from the perspective of V , via namesand the forcing relation.We understand V [G] by jumping inside: “Argue in V [G]”We understand V [G] by analyzing Aut(B).




Set theory – geometry

Geometers reason about the various geometries:externally, as embedded spaces.internally, by using newly formed intuitions.abstractly, using isometry groups.

Similar modes of reasoning arise with a forcing extension V [G]:

We understand V [G] from the perspective of V , via namesand the forcing relation.We understand V [G] by jumping inside: “Argue in V [G]”We understand V [G] by analyzing Aut(B).




Mathematical History

Surely mathematics has a long history of puzzling imaginaryobjects coming to be realized as real.

Irrational numbers√

2 were accepted.0 became a number.Negative numbers became allowed.Complex numbers.Non-Euclidean geometry.

Perhaps it is time for V -generic ultrafilters...




Universe view simulated inside Multiverse

The Universe view is simulated inside the Multiverse by fixing aparticular universe V and declaring it to be the absoluteset-theoretical background, restricting the multiverse to theworlds below V .

This in effect provides absolute background notions ofcountability, well-foundedness, etc., and there are no V -genericfilters in the restricted multiverse...

From the Multiverse perspective, the Universe view appears tobe just like this, restricting attention to a fixed lower cone in themultiverse.




Multiverse adjudicating Universe arguments

The multiverse view seems capable of adjudicating theUniverse view discussion.

Disagreements about the final answers to set-theoreticquestions amount to: which universe V to fix as the Universe?

This is a discussion that can sensibly take place within themultiverse view.



Case study: Multiverse view of the Continuum Hypothesis CH

Case study: the Continuum Hypothesis

The Continuum Hypothesis (CH) is the assertion that every setof reals is either countable or equinumerous with R.

This was a major open question from the time of Cantor, andappeared at the top of Hilbert’s famous list of open problems atthe dawn of the 20th century.

The Continuum Hypothesis is now known to be neitherprovable nor refutable from the usual ZFC axioms of set theory.

Gödel proved that CH holds in the constructible universe L.

Cohen proved that L has a forcing extension L[G] with ¬CH.




CH in the Multiverse

More important than mere independence, both CH and ¬CHare forceable over any model of set theory. Every V has:

V [~c], collapsing no cardinals, such that V [~c] |= ¬CH.V [G], adding no new reals, such that V [G] |= CH.

That is, both CH and ¬CH are easily forceable. We can turn CHon and off like a lightswitch.

We have a deep understanding of how CH can hold and fail,densely in the multiverse, and we have a rich experience in theresulting models. We know, in a highly detailed manner,whether one can obtain CH or ¬CH over any model of settheory, while preserving any number of other features of themodel.




The CH is settled

The multiverse perspective is that the CH question is settled.

The answer consists of our detailed understanding of how theCH both holds and fails throughout the multiverse, of how thesemodels are connected and how one may reach them from eachother while preserving or omitting certain features.

Fascinating open questions about CH remain, of course, but themost important essential facts are known.

In particular, I shall argue that the CH can no longer be settledin the manner that set theorists formerly hoped it might be.




Traditional Dream solution for settling CH

Set theorists traditionally hoped to settle CH this way:

Step 1. Produce a set-theoretic assertion Φ expressing anatural ‘obviously true’ set-theoretic principle. (e.g. AC)

Step 2. Prove that Φ determines CH.That is, prove that Φ→ CH,or prove that Φ→ ¬CH.

And so, CH would be settled, since everyone would accept Φand its consequences.




Dream solution will never be realized

I argue that this template is now unworkable.

The reason is that because of our rich experience andfamiliarity with models having CH and ¬CH, the mere fact of Φdeciding CH immediately casts doubt on its naturality. So wecannot accept such a Φ as obviously true.

In other words, success in the second step exactly underminesthe first step.

Let me present two examples illustrating how this plays out.




Freiling: “a simple philosophical ‘proof’ of ¬CH”

The Axiom of Symmetry (Freiling JSL, 1986)

Asserts that for any function f mapping reals to countable setsof reals, there are x , y with y /∈ f (x) and x /∈ f (y).

Freiling justifies the axiom on pre-theoretic grounds, withthought experiments throwing darts. The first lands at x , soalmost all y have y /∈ f (x). By symmetry, x /∈ f (y).

“Actually [the axiom], being weaker than our intuition,does not say that the two darts have to do anything.All it claims is that what heuristically will happen everytime, can happen.”

Thus, Freiling carries out step 1 in the template.




Freiling carries out step 2

Theorem (Freiling)

The axiom of symmetry is equivalent to ¬CH.

Proof.

If CH, let f (r) be the set of predecessors of r under a fixedwell-ordering of type ω1. So x ∈ f (y) or y ∈ f (x) by linearity.If ¬CH, then for any ω1 many xα, there must be y /∈

⋃α f (xα),

but f (y) contains at most countably many xα.

Thus, Freiling exactly carries out the template.

Was his proposal received as a solution of CH? No.




Objections to Symmetry

Many mathematicians, ignoring Freiling’s pre-reflective appeal,objected from a perspective of deep experience withnon-measurable sets and functions, including extremeviolations of Fubini. For them, the pre-reflective argumentssimply fell flat.

We have become skeptical of naive uses of measure preciselybecause we know the pitfalls; we know how badly behaved setsand functions can be with respect to measure concepts.

Because of our detailed experience, we are not convinced thatAS is intuitively true. Thus, the reception follows my prediction.

And similarly for other dream solutions of CH.




Another example using the dream template

Consider the following set-theoretic principle:

The powerset size axiom PSA

PSA asserts that whenever a set is strictly larger than anotherin cardinality, then it also has strictly more subsets:

∀x , y |x | < |y | ⇒ |P(x)| < |P(y)|.

Set-theorists understand this axiom very well.




Powerset size axiom: |x | < |y | ⇒ |P(x)| < |P(y)|How is this axiom received in non-logic mathematical circles?

Extremely well!

To many mathematicians, this principle is Obvious, as naturaland appealing as AC. Many are surprised to learn it is not atheorem. (Ask your colleagues!)

Meanwhile, set theorists do not agree. Why not? In part,because we know how to achieve all kinds of crazy patternsκ 7→ 2κ via Easton’s theorem. Cohen’s ¬CH model violates it;Martin’s axiom violates it; Luzin’s hypothesis violates it. PSAfails under many of the axioms, such as PFA, MM that are oftenfavored particularly by set-theorists with the universe view.




Powerset size axiom: |x | < |y | ⇒ |P(x)| < |P(y)|How is this axiom received in non-logic mathematical circles?

Extremely well!

To many mathematicians, this principle is Obvious, as naturaland appealing as AC. Many are surprised to learn it is not atheorem. (Ask your colleagues!)

Meanwhile, set theorists do not agree. Why not? In part,because we know how to achieve all kinds of crazy patternsκ 7→ 2κ via Easton’s theorem. Cohen’s ¬CH model violates it;Martin’s axiom violates it; Luzin’s hypothesis violates it. PSAfails under many of the axioms, such as PFA, MM that are oftenfavored particularly by set-theorists with the universe view.




Powerset size axiomSo we have a set-theoretic principle

which many mathematicians find to be obviously true;which expresses an intuitively clear pre-reflective principleabout the concept of size;which set-theorists know is safe and (relatively) consistent;

is almost universally rejected by set-theorists when proposedas a fundamental axiom.

We are too familiar with the ways that PSA can fail, and havetoo much experience working in models where it fails.

But imagine an alternative history, in which PSA is used tosettle a prominent early question and is subsequently adoptedas a fundamental axiom.




Similarly, with CH

I claim that the dream template will not settle CH, because assoon as we know that a proposed principle Φ implies CH orimplies ¬CH, we cannot accept Φ as obvious.

Conclusion. The dream template is unworkable.

We simply have too much experience in and familiarity with theCH and ¬CH worlds. We therefore understand deeply how Φcan fail in worlds that seem perfectly acceptableset-theoretically.




Other attempts to settle CHMore sophisticated attempts to settle CH do not rely on thistraditional template.

Woodin has advanced arguments to settle CH based onΩ-logic, and based on Ultimate-L.

To the extent that an argument aims to settle CH, what theMultiversist desires is an explicit explanation of how ourexperience in the CH or in the ¬CH worlds was somehowillusory, as it seems it must be for the argument to succeed.

Since we have an informed, deep understanding of how it couldbe that CH holds or fails, even in worlds close to any givenworld, it is difficult to regard these worlds as imaginary.



Case Study: Multiverse view of Axiom of Constructibility V = L

The Axiom of Constructibility V = L

The constructible universe L, introduced by Gödel, is built up ina highly regular hierarchy Lα, in which each level consists of thesets explicitly constructed over the previous level.

As a result of this uniform structure, L exhibits numerousattractive regularity features, including the Axiom of Choice andthe Generalized Continuum Hypothesis. The axiom V = Lsettles many questions in set theory.

Indeed, for many of these principles, this was how set-theoristsfirst knew that they were relatively consistent.




Rejecting V = L

Nevertheless, most set theorists reject V = L as a fundamentalaxiom.

V = L is viewed as limiting—why should we have onlyconstructible sets?L can have none of the larger large cardinals.Furthermore, even under V 6= L we can still analyze whatlife would be like in L by the translation ϕ 7→ ϕL.




The multiverse view of L

Nevertheless, the universe L is intensively studied, along withmany other L-like universes accommodating larger largecardinals.

This practice exactly illustrates the multiverse perspective. Wedo not assume V = L as a fundamental axiom, butnevertheless study L as one of many possible set-theoreticuniverses. This is particularly interesting when knowledgeabout L or the other canonical inner models sheds light on V .

The multiverse position also observes that there is not just oneL.




L is not impoverishedObservation

V and L have transitive models of the same theories.

For a given theory in L, the assertion is Σ12, hence absolute.

Observation

Every real exists in a model of ZFC∗ + V = L, well-founded toany desired countable ordinal height.

Statement is true in L and Π12, hence true. Full ZFC is possible if

L has models of ZFC unbounded in ωL1.

True 0] can exist unrecognized in a modelM |= ZFC + V = L, well-founded far beyond ωL

1.Force to collapse ω1 to ω by g. In V [g], there isM |= ZFC + V = L with g ∈ M, well-founded beyond ωV

1 .The multiverse perspective, Harvard EFI 2011 Joel David Hamkins, New York



L is not impoverishedObservation

V and L have transitive models of the same theories.

For a given theory in L, the assertion is Σ12, hence absolute.

Observation

Every real exists in a model of ZFC∗ + V = L, well-founded toany desired countable ordinal height.

Statement is true in L and Π12, hence true. Full ZFC is possible if

L has models of ZFC unbounded in ωL1.

True 0] can exist unrecognized in a modelM |= ZFC + V = L, well-founded far beyond ωL

1.Force to collapse ω1 to ω by g. In V [g], there isM |= ZFC + V = L with g ∈ M, well-founded beyond ωV

1 .The multiverse perspective, Harvard EFI 2011 Joel David Hamkins, New York



Continuing a model to V = L

More generally,

Observation

Every transitive model of set theory can in principle becontinued to a model of V = L.

First force it to be countable; then apply previous.

Perhaps our entire current universe—large cardinals and all—isa transitive model inside a much larger model of V = L.

From this perspective, the models of V = L do not seemparticularly impoverished.



Multiverse Axioms

Somewhat more speculatively, I would like now to proposeprinciples expressing a fuller version of the multiverseperspective.

We seek principles expressing the universe existenceproperties we expect to find in the multiverse.

I aim here to be a little provocative.



Naive multiverse background

The principal multiverse idea is that there should be a largecollection of universes, each a model of (some kind of) settheory.

But we need not consider all universes in the multiverseequally, and we may simply be more interested in the parts ofthe multiverse consisting of universes satisfying very strongtheories, such as ZFC plus large cardinals.

There is little need to draw sharp boundaries, and we mayeasily regard some universes as more set-theoretic than others.



Multiverse as second-order MaximizeThe Multiverse view is expansive.

The multiverse is as big as we can imagine.At any time, we are living inside one of the universes.We do not expect individual worlds to access the wholemultiverse.

Maddy used the MAXIMIZE principle—no undue limitations onset existence—to explain resistance to V = L as an axiom. Itseems unduly limiting that every set should be constructible.

Similarly, here, we follow a second-order version of MAXIMIZE:we seek to place no undue limitations on which universes existin the multiverse.



Universe Existence PrinciplesLet’s begin with some basic multiverse principles.

Realizability Axiom

For any universe V , if W is definable in V and a model of settheory, then W is a universe.

This includes all definable inner models, such as L, HOD, L(R),etc., but also ultrapowers, and so on.

Forcing Extension Axiom

For any universe V and any forcing notion P, there is a forcingextension V [G].

But we also expect non-forcing extensions.



Mirage of tallness

Every model of set theory imagines itself as very tall, containingall the ordinals.

But of course, a model contains only all of its own ordinals, andother models may not agree that these are very tall.

Top Extension Axiom

For every universe V , there is a much taller universe W withV -Wθ.



Mirage of uncountabilityEvery model of set theory imagines itself as enormous,containing the reals R, all the ordinals, etc. It certainly believesitself to be uncountable.

But of course, we know that models can be mistaken about this.There can be countable models of set theory, which do notrealize there is a much larger universe surrounding them.

Countability is a notion that depends on the set-theoreticbackground. On the multiverse perspective, the notion of anabsolute standard of countability falls away.

Countability Axiom

Every universe V is countable from the perspective of another,better universe W .



Mirage of uncountabilityEvery model of set theory imagines itself as enormous,containing the reals R, all the ordinals, etc. It certainly believesitself to be uncountable.

But of course, we know that models can be mistaken about this.There can be countable models of set theory, which do notrealize there is a much larger universe surrounding them.

Countability is a notion that depends on the set-theoreticbackground. On the multiverse perspective, the notion of anabsolute standard of countability falls away.

Countability Axiom

Every universe V is countable from the perspective of another,better universe W .



Absolute standard of Well-foundedness

Set-theorists have traditionally emphasized the well-foundedmodels of set theory.

The concept of well-foundedness, however, depends on theset-theoretic background. Different models of set theory candisagree on whether a structure is well-founded.

Every model of set theory finds its own ordinals to bewell-founded. But we know that models can be incorrect aboutthis.

And this issue arises even with the natural numbers.



The mirage of well-foundedness

On the multiverse view, the idea of a fixed absolute notion ofwell-foundedness falls away.

Well-foundednss Mirage Axiom

Every universe V is ill-founded from the perspective of another,better universe.

In sum, every universe V is a countable ill-founded model fromthe perspective of another, better universe.



Reverse UltrapowersWhen we are living in a universe V and have an ultrapowerembedding V → M, then we can iterate this map forwards

V → M → M2 → M3 → · · ·The later models Mn do not realize that their maps have alreadybeen iterated many times.

Reverse Embedding Axiom

For every universe V and every embedding j : V → M in V ,there is a universe W and embedding h

Wh- V

j- M

such that j is the iterate of h.

Every embedding has already been iterated many times.The multiverse perspective, Harvard EFI 2011 Joel David Hamkins, New York


Absorption into L

We saw earlier how every countable transitive model of settheory can be continued to a model of V = L.

Absorption into L

Every universe V is a countable transitive model in anotheruniverse W satisfying V = L.

For all we know, our current universe is a countable transitivemodel inside L.



A consistency argument

Victoria Gitman and I have proved that, in suitably formalizedform, these multiverse axioms are consistent.

Theorem (Gitman, Hamkins)

If ZFC is consistent, then the collection of countablecomputably-saturated models of ZFC satisfies all the multiverseaxioms.

For example, every countable computably-saturated model ofZFC is a countable transitive model inside another such model,satisfying V = L. And every countable computably-saturated Mhas another such model N in which M is seen to be ill-founded.

So the multiverse vision is not incoherent.



The Multiverse view

On the multiverse view, the task of set theory is not the searchfor the one final True set theory, but rather it is to explore all thevarious interesting set theories that we know of, to find newones, and to discover how they are related.

The philosophical debate will likely not be settled bymathematical proof. But a philosophical view suggestsmathematical questions and topics, and one measure of it isthe value of the mathematics to which it has led.

The philosophical debate may be a proxy for: where should settheory go? Which mathematical questions should it consider?



Mathematical Philosophy

Let me briefly describe some recent mathematics, undertakenfrom the multiverse perspective.

Modal Logic of forcing. Upward-oriented, looking from amodel of set theory to its forcing extensions.Set-theoretic geology. Downward-oriented, looking from amodel of set theory down to its ground models.

This analysis engages pleasantly with various philosophicalviews on the nature of mathematical existence.



Affinity of Forcing & Modal Logic

Since a ground model has access, via names and the forcingrelation, to the objects and truths of the forcing extension, themultiverse exhibits a natural modal nature.

A sentence ϕ is possible or forceable, written ♦ϕ, when itholds in a forcing extension.A sentence ϕ is necessary, written ϕ, when it holds in allforcing extensions.

Many set theorists habitually operate within the correspondingKripke model, even if they wouldn’t describe it that way.



Maximality Principle

The Maximality Principle is the scheme expressing

♦ϕ→ ϕ

Theorem (Hamkins, also Stavi, Väänänen, independently)

The Maximality Principle is relatively consistent with ZFC.

This work led to a consideration: what are the correct modalprinciples of forcing?



Easy forcing validities

K (ϕ→ ψ)→ (ϕ→ ψ)

Dual ¬ϕ↔ ¬♦ϕS ϕ→ ϕ

4 ϕ→ ϕ.2 ♦ϕ→ ♦ϕ

Theorem

Any S4.2 modal assertion is a valid principle of forcing.

Question

What are the valid principles of forcing?




K (ϕ→ ψ)→ (ϕ→ ψ)


4 ϕ→ ϕ.2 ♦ϕ→ ♦ϕ

Theorem


Question





K (ϕ→ ψ)→ (ϕ→ ψ)


4 ϕ→ ϕ.2 ♦ϕ→ ♦ϕ

Theorem


Question




Beyond S4.2

5 ♦ϕ→ ϕM ♦ϕ→ ♦ϕ

W5 ♦ϕ→ (ϕ→ ϕ).3 ♦ϕ ∧ ♦ψ → (♦(ϕ ∧ ♦ψ) ∨ ♦(ϕ ∧ ψ) ∨ ♦(ψ ∧ ♦ϕ))

Dm ((ϕ→ ϕ)→ ϕ)→ (♦ϕ→ ϕ)Grz ((ϕ→ ϕ)→ ϕ)→ ϕLöb (ϕ→ ϕ)→ ϕ

H ϕ→ (♦ϕ→ ϕ)

It is a fun forcing exercise to show that these are invalid in some or allmodels of ZFC.



The resulting modal theories

S5 = S4 + 5S4W5 = S4 + W5

S4.3 = S4 + .3S4.2.1 = S4 + .2 + M

S4.2 = S4 + .2S4.1 = S4 + M

S4 = K4 + SDm.2 = S4.2 + Dm

Dm = S4 + DmGrz = K + GrzGL = K4 + Löb

K4H = K4 + HK4 = K + 4K = K + Dual

S5

S4W5?

S4.2.1 S4.3?

Dm.2

-Grz

S4.1?

S4.2?

-Dm?

K4H

GL

S4?

-

K4?

-

K?



Valid principles of forcing

Theorem (Hamkins,Löwe)

If ZFC is consistent, then the ZFC-provably valid principles offorcing are exactly S4.2.

It was easy to show that S4.2 is valid.

The difficult part is to show that nothing else is valid.

If S4.2 6` ϕ, we must provide ψi such that ϕ(ψ0, . . . , ψn) fails insome model of set theory.



Buttons and Switches

A switch is a statement ϕ such that both ϕ and ¬ϕ arenecessarily possible.A button is a statement ϕ such that ϕ is (necessarily)possibly necessary.

Fact. Every statement in set theory is either a switch, a buttonor the negation of a button.

Theorem

If V = L, then there is an infinite independent family of buttonsand switches.



Structure of the argumentIn order to show that non-S4.2 assertions are also not validprinciples of forcing, we proceeded in two steps:

First, we proved a new completeness theorem for S4.2frames: the class of finite pre-lattices.Second, we proved that any finite Kripke model on such aframe is simulated via Boolean combinations of buttonsand switches by forcing over a fixed model of set theory.

Step 2 made essential use of the Jankov-Fine modalexpressions.

Theorem (Hamkins, Löwe)

If ZFC is consistent, then the ZFC-provable forcing validities areexactly S4.2.



Structure of the argumentIn order to show that non-S4.2 assertions are also not validprinciples of forcing, we proceeded in two steps:

First, we proved a new completeness theorem for S4.2frames: the class of finite pre-lattices.Second, we proved that any finite Kripke model on such aframe is simulated via Boolean combinations of buttonsand switches by forcing over a fixed model of set theory.

Step 2 made essential use of the Jankov-Fine modalexpressions.

Theorem (Hamkins, Löwe)

If ZFC is consistent, then the ZFC-provable forcing validities areexactly S4.2.



Validities in a modelA fixed model W of set theory can exhibit more than just theprovable validities. Nevertheless,

Theorem

S4.2 ⊆ ForceW ⊆ S5.

Both endpoints occur for various W .

Questions

1 Can the validities of W be strictly intermediate?2 If ϕ is valid for forcing over W , does it remain valid for

forcing over all extensions of W?3 Can a model of ZFC have an unpushed button, but not two

independent buttons?The multiverse perspective, Harvard EFI 2011 Joel David Hamkins, New York


Surprising entry of large cardinalsTheorem. The following are equiconsistent:

1 S5(R) is valid.2 S4W5(R) is valid for forcing.3 Dm(R) is valid for forcing.4 There is a stationary proper class of inaccessible cardinals.

Theorem

1 (Welch,Woodin) If S5(R) is valid in all forcing extensions(using R of extension), then ADL(R).

2 (Woodin) If ADR + Θ is regular, then it is consistent thatS5(R) is valid in all extensions.



Set-theoretic geology: A new perspective

Forcing is naturally viewed as a method of building outer asopposed to inner models of set theory.

Nevertheless, a simple switch in perspective allows us to viewforcing as a method of producing inner models as well.

Namely, we look for how the universe V might itself have arisenvia forcing. Given V , we look for an inner model W over whichthe universe is a forcing extension:

W ⊆W [G] = V

This change in viewpoint results in the subject we callset-theoretic geology.



Set-theoretic geology: A new perspective

Forcing is naturally viewed as a method of building outer asopposed to inner models of set theory.

Nevertheless, a simple switch in perspective allows us to viewforcing as a method of producing inner models as well.

Namely, we look for how the universe V might itself have arisenvia forcing. Given V , we look for an inner model W over whichthe universe is a forcing extension:

W ⊆W [G] = V

This change in viewpoint results in the subject we callset-theoretic geology.



Digging for GroundsA ground of the universe V is a class W over which theuniverse arises by forcing V = W [G].

Theorem (Laver, independently Woodin)

Every ground W is a definable class in its forcing extensionsW [G], using parameters in W.

Definition (Hamkins, Reitz)

The Ground Axiom is the assertion that the universe V has nonontrivial grounds.

Theorem (Reitz)

The Ground Axiom is first order expressible in set theory.



Digging for GroundsA ground of the universe V is a class W over which theuniverse arises by forcing V = W [G].

Theorem (Laver, independently Woodin)

Every ground W is a definable class in its forcing extensionsW [G], using parameters in W.

Definition (Hamkins, Reitz)

The Ground Axiom is the assertion that the universe V has nonontrivial grounds.

Theorem (Reitz)

The Ground Axiom is first order expressible in set theory.



The Ground AxiomThe Ground Axiom holds in many canonical models of settheory: L, L[0]], L[µ], many instances of K .

Question

To what extent are the highly regular features of these modelsconsequences of GA?

Theorem (Reitz)

Not at all. Every model of ZFC has an extension, preservingany desired Vα, which is a model of GA.

Theorem (Hamkins, Reitz, Woodin)

Every model of ZFC has an extension, preserving any desiredVα, which is a model of GA + V 6= HOD.



BedrocksW is a bedrock of V if it is a ground of V and minimal withrespect to the forcing extension relation.

Equivalently, W is a bedrock of V if it is a ground of V andsatisfies GA.

Open Question

Is the bedrock unique when it exists?

Theorem (Reitz)

It is relatively consistent with ZFC that the universe V has nobedrock model.

Such models are bottomless.The multiverse perspective, Harvard EFI 2011 Joel David Hamkins, New York


The principle new concept of geology: the mantleDefinition

The Mantle M is the intersection of all grounds.

The analysis engages with an interesting philosophical view:

Ancient Paradise. This is the philosophical view that there is ahighly regular core underlying the universe of set theory, aninner model obscured over the eons by the accumulating layersof debris heaped up by innumerable forcing constructions sincethe beginning of time. If we could sweep the accumulatedmaterial away, we should find an ancient paradise.

The Mantle, of course, wipes away an entire strata of forcing.

Haim Gaifman has pointed out (with humor) that the geologyterminology presumes a stand on whether forcing is a humanactivity or a natural one...



The principle new concept of geology: the mantleDefinition

The Mantle M is the intersection of all grounds.

The analysis engages with an interesting philosophical view:

Ancient Paradise. This is the philosophical view that there is ahighly regular core underlying the universe of set theory, aninner model obscured over the eons by the accumulating layersof debris heaped up by innumerable forcing constructions sincethe beginning of time. If we could sweep the accumulatedmaterial away, we should find an ancient paradise.

The Mantle, of course, wipes away an entire strata of forcing.

Haim Gaifman has pointed out (with humor) that the geologyterminology presumes a stand on whether forcing is a humanactivity or a natural one...



Every model is a mantle

Although the Ancient Paradise philosophical view is highlyappealing, our main theorem tends to refute it.

Main Theorem (Fuchs, Hamkins, Reitz)

Every model of ZFC is the mantle of another model of ZFC.

By sweeping away the accumulated sands of forcing, what wefind is not a highly regular ancient core, but rather: an arbitrarymodel of set theory.



Downward directednessThe grounds of V are downward directed if the intersection ofany two of them contains a third.

Wt ⊆Wr ∩Ws

In this case, there could not be two distinct bedrocks.

Open Question

Are the grounds downward directed? Downward set directed?

This is a fundamental open question about the nature of themultiverse.

In every model for which we have determined the answer, theanswer is yes.



The Mantle under directedness

Theorem

1 If the grounds are downward directed, then the Mantle isconstant across the grounds, and M |= ZF.

2 If they are downward set-directed, then M |= ZFC.

The Generic Mantle, denoted gM, is the intersection of allgrounds of all forcing extensions of V . This includes all thegrounds of V , and so gM ⊆ M.

Theorem

If the generic grounds are downward directed, then the groundsare dense below the generic multiverse, and so M = gM.



The Mantle under directedness

Theorem

1 If the grounds are downward directed, then the Mantle isconstant across the grounds, and M |= ZF.

2 If they are downward set-directed, then M |= ZFC.

The Generic Mantle, denoted gM, is the intersection of allgrounds of all forcing extensions of V . This includes all thegrounds of V , and so gM ⊆ M.

Theorem

If the generic grounds are downward directed, then the groundsare dense below the generic multiverse, and so M = gM.



The generic multiverseThe Generic Multiverse of a model of set theory is the part ofthe multiverse reachable by forcing, the family of universesobtained by closing under forcing extensions and grounds.

There are various philosophical motivations to study thegeneric multiverse.

Woodin introduced the generic multiverse in order to reject acertain multiverse view of truth, the view that a statement is truewhen it is true in every model of the generic multiverse.

Our view is that the generic multiverse is a fundamental featureand should be a major focus of study.

The generic multiverse of a model is the local neighborhood ofthat model in the multiverse.



The generic multiverseThe Generic Multiverse of a model of set theory is the part ofthe multiverse reachable by forcing, the family of universesobtained by closing under forcing extensions and grounds.

There are various philosophical motivations to study thegeneric multiverse.

Woodin introduced the generic multiverse in order to reject acertain multiverse view of truth, the view that a statement is truewhen it is true in every model of the generic multiverse.

Our view is that the generic multiverse is a fundamental featureand should be a major focus of study.

The generic multiverse of a model is the local neighborhood ofthat model in the multiverse.



The Generic MantleThe generic Mantle is tightly connected with the genericmultiverse.

Theorem

The generic Mantle gM is invariant by forcing and gM |= ZF.

Corollary

The generic Mantle gM is constant across the genericmultiverse. In fact, gM is the intersection of the genericmultiverse. It is the largest forcing-invariant class.

On this view, the generic Mantle is a canonical, fundamentalfeature of the generic multiverse.



The Generic HODHOD is the class of hereditarily ordinal definable sets.

HOD |= ZFC

The generic HOD, introduced by Fuchs, is the intersection of allHODs of all forcing extensions.

gHOD =⋂G

HODV [G]

gHOD is constant across the generic multiverse.The HODV [G] are downward set-directed. gHOD |= ZFC.

HOD∪

gHOD ⊆ gM ⊆ M



The Generic HODHOD is the class of hereditarily ordinal definable sets.

HOD |= ZFC

The generic HOD, introduced by Fuchs, is the intersection of allHODs of all forcing extensions.

gHOD =⋂G

HODV [G]

gHOD is constant across the generic multiverse.The HODV [G] are downward set-directed. gHOD |= ZFC.

HOD∪

gHOD ⊆ gM ⊆ M



Realizing V as the Mantle

Theorem (Fuchs, Hamkins, Reitz)

If V |= ZFC, then there is a class extension V in which

V = MV = gMV = gHODV = HODV

In particular, as mentioned earlier, every model of ZFC is themantle and generic mantle of another model of ZFC.

It follows that we cannot expect to prove ANY regularityfeatures about the mantle or the generic mantle.



Theorem (Fuchs, Hamkins, Reitz)

Other combinations are also possible.1 Every model of set theory V has an extension V with

V = MV = gMV = gHODV = HODV

2 Every model of set theory V has an extension W with

V = MW = gMW = gHODW but HODW = W

3 Every model of set theory V has an extension U with

V = HODU = gHODU but MU = U

4 Lastly, every V has an extension Y with

Y = HODY = gHODY = MY = gMY



Thank you.

Joel David HamkinsThe City University of New York

http://jdh.hamkins.org


Documents

The multiverse perspective on determinateness in set theorylogic.harvard.edu/EFI_Hamkins_MultiverseSlides.pdf · The multiverse view Ontology of Forcing Case Studies: CH, V = L Multiverse