Logical Analysis of Ramsey’s Theorem - SJTU · Logical Analysis of Ramsey’s Theorem ... Mathematical induction: If P is a property about numbers ... (Rogers, Theory of Recursive

Nonstandard Models and ReverseMathematics:

Logical Analysis of Ramsey’s Theorem

Chong Chi Tat

National University of Singapore

[email protected]

24 August 2014

Peano Arithmetic PA

Basic rules of arithmetic operation, such as∀x∀y(x + y = y + x) etc.∀x(x ≠ 0→ ∃y(x = y + 1)Mathematical induction: If P is a property about numbersexpressed in the language of PA, such that P(0) andP(n + 1) whenever P(n), then

∀xP(x).

Fundamental Question. Does PA completely characterizethe set N of natural numbers?What is “truth” in mathematics?

Peano Arithmetic PA


∀xP(x).


Peano Arithmetic PA


∀xP(x).


Peano Arithmetic PA


∀xP(x).


Peano Arithmetic PA


∀xP(x).


Peano Arithmetic PA


∀xP(x).


Nonstandard Models of Arithmetic

Gödel’s Incompleteness Theorem. There is a statement ϕabout “numbers” in the language of first order arithmeticthat is not provable.“Provable” = “Provable within the system PA”There are two models of PA: ϕ is true in one and false inthe other.One of the two models is not the standard model⟨N,+, ⋅,0,1⟩.That model is a nonstandard model of arithmetic.









Nonstandard Models of PA

Skolem (1934): Explicit construction of anonstandard model.In a nonstandard model of PA, there are “standardnumbers” 0,1, . . . ,n, . . . . and “nonstandard numbers”. . . ,a − 1,a,a + 1, . . .

N is an “initial segment” of the set of numbers in the model.“n is standard” is not definable in the language of PA.N is a typical example of a cut in a nonstandard model.










Nonstandard Models After 1934

Applications to classical mathematics: (AbrahamRobinson) Nonstandard analysisInvestigations of models of PA (various authors)Computation theory over nonstandard models (since1980’s)







The Heart of Computation

What lies at the heart of a theory of computation?“Effective computability” (Rogers, Theory of RecursiveFunctions and Effective Computability)Effective computability = Algorithms = Turing machines= Definable by a “∆0

1 formula” in PA



1 formula” in PA



1 formula” in PA



1 formula” in PA



1 formula” in PA

Formalizing the Notion of Effective Computability

f is partial recursive iff there is a Σ01 formula ∃xϕ such that

f (m) = n⇔ PA ⊢ ∃xϕ(x ,m,n).

(*) Every partial recursive function maps a finite set onto afinite set. [Trivial for N but not so in general.]X ⊆ N is recursively enumerable (r.e.) iff there is a Σ0

1formula ∃xϕ such that

n ∈ X ⇔ PA ⊢ ∃xϕ(x ,n).

X ≤T Y (Turing reducible) iff X and X̄ are Σ01 definable in Y .



f (m) = n⇔ PA ⊢ ∃xϕ(x ,m,n).



n ∈ X ⇔ PA ⊢ ∃xϕ(x ,n).




f (m) = n⇔ PA ⊢ ∃xϕ(x ,m,n).



n ∈ X ⇔ PA ⊢ ∃xϕ(x ,n).




f (m) = n⇔ PA ⊢ ∃xϕ(x ,m,n).



n ∈ X ⇔ PA ⊢ ∃xϕ(x ,n).




f (m) = n⇔ PA ⊢ ∃xϕ(x ,m,n).



n ∈ X ⇔ PA ⊢ ∃xϕ(x ,n).


Definability

In general, given X ⊆ N,Investigate computational aspects of sets Y definable in X :

y ∈ Y ⇔ Qx1Qx2⋯Qxnϕ(X ,x1, . . . ,xn,y),

where Q = ∃ or ∀ and ϕ is ∆00. [Y is “Σ0

n or Π0n definable” in

X .]

Thus

Recursion Theory = Theory of Definability

Definability





X .]

Thus


Definability





X .]

Thus


Generalizing Recursion Theory

Question. Can one develop recursion theory over adomain different from N?Second order arithmetic (Kleene, Kreisel, Spector 1960’s):X ⊂ N is “r.e.” iff X is “Π1

1”—

n ∈ X ⇔ ∀Y∃xϕ(Y ↾ x ,n)⇔ LωCK

1⊧ ∃xψ(x ,n),

where ϕ is a ∆00 formula in arithmetic, ∃xψ is a Σ1 formula

in set theory, and LωCK1

is Gödel’s constructible universe upto Church-Kleene ω1.



1”—


1⊧ ∃xψ(x ,n),






1”—


1⊧ ∃xψ(x ,n),






1”—


1⊧ ∃xψ(x ,n),





Gödel’s L (Sacks 1960–70’s): An ordinal α is admissible ifLα supports a theory of computation.f is partial recursive if it is Σ1 definable over Lα.Lα supports a recursion theory if every partial recursivefunction maps a “finite set” in Lα onto a “finite set” in Lα.It is possible to introduce the notion of a Turing machine inLα.One can define “relative computability” and degree ofunsolvability.











Gödel’s Constructible Universe L

��

LLLLLLLLLLLLLLLLLLL

∅

rω

r αr ωCK1

Recursion Theory in a Model M of PA

Full induction is NOT at the heart of computability.Σ0

1 definability captures the notion of “effectivecomputability”.A function is partial recursive if it is Σ0

1 definable in M.BΣ0

1: Every partial recursive function maps a “finite set” inM onto a “finite set”.One can develop a theory of computation for anonstandard model M of BΣ0

1.The existence of “cuts” presents a whole new picture.For example, what are the recursion-theoretic properties ofa cut?





































Fragments of PA

Let P− = PA ∖ Induction.(Paris and Kirby 1978): Over P−, there is a hierarchy ofsubsystems of PA of increasing complexity and strength:

⋅ ⋅ ⋅→ BΣ0n+1 → IΣ0

n → BΣ0n → . . .

→ IΣ02 → BΣ0

2 → IΣ01 → BΣ0

1

IΣ0n: Induction for Σ0

n formulasBΣ0

n: Every Σ0n definable function maps a “finite set” onto a

“finite set”.

Fragments of PA


⋅ ⋅ ⋅→ BΣ0n+1 → IΣ0

n → BΣ0n → . . .

→ IΣ02 → BΣ0

2 → IΣ01 → BΣ0

1


n formulasBΣ0


“finite set”.

Fragments of PA


⋅ ⋅ ⋅→ BΣ0n+1 → IΣ0

n → BΣ0n → . . .

→ IΣ02 → BΣ0

2 → IΣ01 → BΣ0

1


n formulasBΣ0


“finite set”.

Fragments of PA


⋅ ⋅ ⋅→ BΣ0n+1 → IΣ0

n → BΣ0n → . . .

→ IΣ02 → BΣ0

2 → IΣ01 → BΣ0

1


n formulasBΣ0


“finite set”.

Σ0n-Functions on a Cut

Without Σ0n-induction: A Σ0

n-function f on a cut with unboundedimage.

PART II

Reverse Mathematics:From First Order to Second Order Arithmetic

The Base Theory RCA0

RCA0 (H Friedman: Recursive Comprehension Axiom) consistsof

P−

“If X is a set in M, and Y ≤T X , then Y is also in M.”For each X ∈M, mathematical induction forΣ0

1(X)-formulas ϕ:

(ϕ(0) ∧ ∀x(ϕ(x)→ ϕ(x + 1))→ ∀xϕ(x)

Models of RCA0 include both standard and nonstandardmodels of P− + IΣ0

1.



P−


1(X)-formulas ϕ:

(ϕ(0) ∧ ∀x(ϕ(x)→ ϕ(x + 1))→ ∀xϕ(x)


1.



P−


1(X)-formulas ϕ:

(ϕ(0) ∧ ∀x(ϕ(x)→ ϕ(x + 1))→ ∀xϕ(x)


1.



P−


1(X)-formulas ϕ:

(ϕ(0) ∧ ∀x(ϕ(x)→ ϕ(x + 1))→ ∀xϕ(x)


1.



P−


1(X)-formulas ϕ:

(ϕ(0) ∧ ∀x(ϕ(x)→ ϕ(x + 1))→ ∀xϕ(x)


1.



P−


1(X)-formulas ϕ:

(ϕ(0) ∧ ∀x(ϕ(x)→ ϕ(x + 1))→ ∀xϕ(x)


1.

Foundational Question

Hilbert’s program: Identify the set existence axioms necessary(and sufficient) to prove theorems in mathematics.

Given a mathematical theorem Φ, find a sentence ϕ aboutset existence such that

RCA0 ⊢ Φ↔ ϕ.

Model-theoretically, this means

For all M,M ⊧ RCA0 ⇒M ⊧ Φ↔ ϕ.




RCA0 ⊢ Φ↔ ϕ.






RCA0 ⊢ Φ↔ ϕ.






RCA0 ⊢ Φ↔ ϕ.



RCA0

(Simpson and many others) There exist (big) five systemsabout set existence that classify many mathematicaltheorems.Since 2007, this orderly pattern has been shown to fail ingeneral, especially for combinatorial principles.Instead, we now have picture of a zoo.

RCA0


RCA0


RCA0


The Big Five Systems

Ramsey’s Theorem

Theorem

(F P Ramsey (1931)) RTnk ∶ Any coloring of the n-element subsets

of N into k colors has an infinite homogeneous set, i.e. an A ⊆ Nall of whose n-element sets have the same color.

RTnk is a theorem of Peano arithmetic for any n,k .

Problem: Analyze the proof-theoretic strength of RTnk (and

related principles) over the base system RCA0.

Ramsey’s Theorem

Theorem

(F P Ramsey (1931)) RTnk ∶ Any coloring of the n-element subsets

of N into k colors has an infinite homogeneous set, i.e. an A ⊆ Nall of whose n-element sets have the same color.

RTnk is a theorem of Peano arithmetic for any n,k .

Problem: Analyze the proof-theoretic strength of RTnk (and

related principles) over the base system RCA0.

Three Problems Related to RTnk

First problem:

Compare the strengths of RTnk for different n,k ’s over

RCA0.

Question. (For example) Is RT32 stronger than RT2

2? Is RT22

stronger than RT12 (the pigeonhole principle)?

Either show every model of RCA0 +RT22 is a model of RT3

2,or exhibit a model of RCA0 +RT2

2 that is not a model of RT32.


First problem:


RCA0.


2? Is RT22






First problem:


RCA0.


2? Is RT22






First problem:


RCA0.


2? Is RT22






First problem:


RCA0.


2? Is RT22





2nd Problem: The SRT22 Principle

A 2-coloring is stable if for all x , (x ,y) has the same color for allsuffciently large y .

SRT22: Every stable two coloring of numbers has a

homogeneous set.Clearly RT2

2 implies SRT22.

Question. Does SRT22 imply RT2

2?





2 implies SRT22.


2?





2 implies SRT22.


2?





2 implies SRT22.


2?





2 implies SRT22.


2?

3rd Problem: First and Second Order Strength

Question. Which of RT22, SRT2

2 implis BΣ02 or IΣ0

2?

3rd Problem: First and Second Order Strength

Question. Which of RT22, SRT2

2 implis BΣ02 or IΣ0

2?

Ramsey’s Theorem for Pairs

What is known: Over RCA0,Jockusch (1972): RTn

k , for n ≥ 3 and k ≥ 2, impliesACA0—Any model of RCA0 +RT3

2 is closed under theTuring jump operation.Seetapun and Slaman (1995): There is a model ofRCA0 +RT2

2 that does not include ∅′.This settled Problem 1.





















First Order Consequence

Each of the following implies BΣ02 over RCA0:

Hirst (1987): RT22

Cholak, Jockusch and Slaman (2001): SRT22



Hirst (1987): RT22




Hirst (1987): RT22




Hirst (1987): RT22


RT22 and SRT2

2

Theorem (Chong, Slaman and Yang (2014))

There is a model M of RCA0 +BΣ02 such that

(1) SRT22 +WKL0 holds but not RT2

2.(2) IΣ0

2 fails.

These answer Problem 2 (separate SRT22 from RT2

2) andhalf of Problem 3.The model M for (1) and (2) above is a nonsrtandardmodel of RCA0 +BΣ0

2.To satisfy (2), M has to be nonstandard.

RT22 and SRT2

2




2.(2) IΣ0

2 fails.




RT22 and SRT2

2




2.(2) IΣ0

2 fails.




RT22 and SRT2

2




2.(2) IΣ0

2 fails.




RT22 and SRT2

2




2.(2) IΣ0

2 fails.




RT22 and SRT2

2




2.(2) IΣ0

2 fails.




RT22 and SRT2

2




2.(2) IΣ0

2 fails.




The Model M

In M, every set is recursive or low.Low set = Its jump is Turing equivalent to ∅′

No standard model with only low sets separates SRT22 from

RT22.

The Model M



RT22.

The Model M



RT22.

The Model M



RT22.

RT22 and BΣ0

2

Over RCA0, does RT22 imply IΣ0

2?

Theorem (Chong, Slaman and Yang (To appear))

There is a model of RCA0 +RT22 + ¬IΣ0

2. Hence RT22 does not

imply Σ02 induction.

Does SRT22 imply RT2

2 in any model of IΣ02, in particular, in

a standard model?

RT22 and BΣ0

2


2?







a standard model?

RT22 and BΣ0

2


2?







a standard model?

RT22 and BΣ0

2


2?







a standard model?

A Zoo Diagram

A Look At the Zoo From Afar...

Nonstandard Models in Reverse Mathematics

Philosophical consideration:What is the “correct” mathematical framework for the studyof reverse mathematics?In particular, is Σ0

1-induction the most appropriate schemefor the conceptualization of mathematical induction, or forinvestigation of combinatorial principles?










References

Chong, C. T., Theodore A. Slaman and Yue Yang, Themetamathematics of stable Ramsey’s theorem for pairs,Journal of American Mathematical Society (2014),863–891Chong, C. T, Wei Li and Yue Yang, Nonstandard models inrecursion theory and reverse mathematics, Bulletin ofSymbolic Logic (2014), to appearChong, C. T, Theodore A. Slaman and Yue Yang, Theinductive strength of Ramsey’s theorem for pairs, preprint

Thank You!

Documents

Logical Analysis of Ramsey’s Theorem - SJTU · Logical Analysis of Ramsey’s Theorem ... Mathematical induction: If P is a property about numbers ... (Rogers, Theory of Recursive