tutorial series on reverse mathematics denis r. …tutorial series on reverse mathematics denis r....

Tutorial Series on Reverse Mathematics

Denis R. Hirschfeldt — University of Chicago

2017 NZMRI Summer School, Napier, New Zealand

Part I: Background

A Bit of Historical Context

A Bit of Historical Context: Abstraction and the Loss of Certainty

Concrete, algorithmic mathematics

19th c.−→ Abstract mathematics

Increase in power, but also a loss of intuition

Increased demand for rigor

Cantor’s Paradise

Russell’s Paradox: Let S = {A : A /∈ A}. Is S ∈ S?

Crisis in foundations

Hilbert’s Program: prove the consistency of mathematics viafinitistic methods

Godel’s Second Incompleteness Theorem

Concrete, algorithmic mathematics 19th c.−→ Abstract mathematics

Cantor’s Paradise

Russell’s Paradox

: Let S = {A : A /∈ A}. Is S ∈ S?

Cantor’s Paradise

A Bit of Historical Context: Reverse Mathematics

Nada se edifica sobre la piedra, todo sobre la arena,pero nuestro deber es edificar como si fuera piedra laarena.

— Jorge Luis Borges

We can still try to understand how much axiomatic power giventheorems need.

Fix a weak base axiomatic system B.

Given a theorem T , we can find an axiomatic system S ⊇ Bsufficient to prove T .

If we can then also show that the axioms of S are provable fromB + T , then we know S is exactly what we need to prove T .

We can also compare theorems in terms of implication over B.

A Bit of Historical Context: Algorithms in Mathematics

Loss of algorithmic content

Increased interest in the notion of computability

Hilbert’s Entscheidungsproblem: algorithm to decide, given a setof axioms A and a statement σ, whether σ follows from A

Ramon Llull (c. 1232–1315), Gottfried Leibniz (1646–1716)

The writing machine at theA Llullian circle Grand Academy of Lagado

(Gulliver’s Travels, 1726)

Emil du Bois-Reymond’s ignoramus et ignorabimus (1880) andHilbert’s “Wir mussen wissen—wir werden wissen.” (1930)

Hilbert’s 10th Problem: algorithm to decide whether a givenDiophantine equation has a solution

A Bit of Historical Context: Computability Theory

Despite Hilbertian optimism, not all problems have algorithms.

Examples require a formal notion of computability.

Various proposed definitions by Church, Godel, Herbrand,Kleene in the 1930’s

Turing’s machine-based definition (1936)

All of these definitions are equivalent.

Church-Turing Thesis: This definition captures the intuitive notionof “computable”.

Thm (Church; Turing). There is no such algorithm.

Thm (Davis; Putnam; Robinson; Matiyasevich). There is no suchalgorithm.

Many other objects have been shown to be noncomputable.

Computability theory has tools to compare such objects.

A Bit of Computability Theory

We look at countably infinite objects built out of finite ones, e.g.sets of natural numbers, sets of finite strings, functions N→ N, etc.

Computability for such objects can bethought of via an informal idea of algorithm;defined formally using a model such as Turing machines.

We can list all Turing machines (with inputs and outputs in N), insuch a way that we can simulate the computation of the eth

machine on input n using a universal Turing machine.

A Turing machine may fail to halt on a given input, so this listyields a list Φ0,Φ1, . . . of all partial computable functions.

We write Φe(n)↓ to mean that Φe is defined on n.

The Halting Problem is ∅′ = {〈e,n〉 : Φe(n)↓}.

Thm (Turing). ∅′ is not computable.

Pf. By diagonalization: Suppose that ∅′ is computable.

Then so is f (e) =

{Φe(e) + 1 if 〈e,e〉 ∈ ∅′

0 otherwise.

Thus Φe = f for some e.

Then Φe(e)↓ = f (e) = Φe(e) + 1. �

A similar proof shows that there is no effective list of all totalcomputable functions.

Then so is f (e) =

{Φe(e) + 1 if 〈e,e〉 ∈ ∅′

0 otherwise.

Then Φe(e)↓ = f (e) = Φe(e) + 1. �

Then so is f (e) =

{Φe(e) + 1 if 〈e,e〉 ∈ ∅′

0 otherwise.

Then Φe(e)↓ = f (e) = Φe(e) + 1. �

Then so is f (e) =

{Φe(e) + 1 if 〈e,e〉 ∈ ∅′

0 otherwise.

Then Φe(e)↓ = f (e) = Φe(e) + 1. �

∅′ is not computable, but it is computably enumerable (c.e.).

So are the sets in the Entscheidungsproblem and in Hilbert’s 10th

problem.

A is computable relative to B if there is an algorithm forcomputing A if given access to B.

Can be formalized using Turing machines with oracle tapes.

We write A 6T B and say that A is Turing reducible to B.

If A 6T B and B 6T A then A and B are Turing equivalent.

The resulting equivalence classes are the Turing degrees.

The degree of the join A⊕ B = {2n : n ∈ A} ∪ {2n + 1 : n ∈ B} is theleast upper bound of the degrees of A and B.

problem.

For a c.e. A, define a partial computable f s.t. f (n)↓ iff n ∈ A.

∅′ can tell whether f (n)↓, so A is computable relative to ∅′.

We say that ∅′ is a complete c.e. set.

The undecidability of the Entscheidungsproblem and of Hilbert’s10th problem are proved by encoding ∅′.

So the corresponding c.e. sets are also complete, i.e., they are inthe same Turing degree as ∅′.

Thm (Friedberg; Muchnik). There are noncomputable,incomplete c.e. sets.

There are also non-c.e. sets that are computable relative to ∅′,including co-c.e. sets but also many others.

We can relativize other computability-theoretic concepts.

We can define the concept of being c.e. relative to X .

Let ΦX0 ,Φ

X1 , . . . be the functions that are partial computable

relative to X .

We can define the Halting Problem relative to X asX ′ = {〈e,n〉 : ΦX

e(n)↓}.

We call this the (Turing) jump of X .

If X 6T Y then X ′ 6T Y ′, but not necessarily vice-versa.

Computability-theoretic results tend to relativize.

E.g., X ′ is not computable relative to X , and is complete for setsc.e. relative to X .

Let ΦX0 ,Φ

relative to X .

e(n)↓}.

Let ΦX0 ,Φ

relative to X .

e(n)↓}.

Let ΦX0 ,Φ

relative to X .

e(n)↓}.

Let ΦX0 ,Φ

relative to X .

e(n)↓}.

Let ΦX0 ,Φ

relative to X .

e(n)↓}.

Part II: Computability-Theoretic Comparison

An Example: Versions of Konig’s Lemma

Trees and Paths

A tree is a subset T of N<ω closed under initial segments.

T is computable if there is an algorithm for determining whethera given σ is in T .

T is finitely branching if for each σ ∈ T , |{n : σn ∈ T}| <∞.

T is binary if it is a subset of 2<ω.

A path on T is a P ∈ Nω s.t. every initial segment of P is in T .

Put a topology on Nω by taking {X : σ ≺ X} as basic open sets.

Then C is closed iff it is the set of paths on a tree.

Put a measure on 2ω by letting µ({X : σ ≺ X}) = 2−|σ|.

Trees and Paths

Versions of Konig’s Lemma

Konig’s Lemma: Every infinite, finitely branching tree has a path.

Weak Konig’s Lemma: Every infinite binary tree has a path.

Weak Weak Konig’s Lemma: Every binary tree T s.t.

lim infn|{σ∈T :|σ|=n}|

2n > 0

has a path.

Bounded Konig’s Lemma: Every infinite binary tree T s.t.

|{σ ∈ T : |σ| = n}| < c

for some c has a path.

2n > 0

has a path.

|{σ ∈ T : |σ| = n}| < c

2n > 0

has a path.

|{σ ∈ T : |σ| = n}| < c

2n > 0

has a path.

|{σ ∈ T : |σ| = n}| < c

KL: Infinite, finitely branching trees have paths.

WKL: Infinite binary trees have paths.

WWKL: Fat binary trees have paths.

BKL: Skinny infinite binary trees have paths.

WKL says that 2ω is compact.

KL says that certain subspaces of Nω are compact, but thesesubspaces are not as effectively presented.

WKL: Find an element of a closed set.

WWKL: Find an element of a closed of positive measure.

BKL: Find an element of a finite set.

Bounded Konig’s Lemma and Computability

Let T be a computable infinite binary tree s.t.|{σ ∈ T : |σ| = n}| < c for all n.

There is a σ ∈ T extended by a unique path P on T .

For each n > |σ|, there is a unique τn � σ of length n s.t. T isinfinite above τn.

We can find τn computably.

P = limn τn, so T has a computable path.

In fact, every path on T is computable.

More generally, even if T is not computable, the aboveprocedure is computable relative to T .

Thus BKL is computably true.

Konig’s Lemma and Computability

Thm (Kreisel). There is a computable infinite binary tree with nocomputable path.

Thus WKL is not computably true, and hence neither is KL.

Kreisel’s tree can be fat, so WWKL is also not computably true.

To build such a tree, we diagonalize against all potentialcomputable paths.

There is a computable infinite, finitely branching tree T s.t. everypath of T computes ∅′.

There is a computable infinite, finitely branching tree T with nopath computable relative to ∅′.

Weak Konig’s Lemma and Computability

Let T be a computable infinite binary tree.

Thm (Kreisel). T has a path P 6T ∅′.

An example is the leftmost path of T .

Thm (Shoenfield). T has a path P <T ∅′.

Thus WKL is strictly weaker than KL in at least two senses.

But just how much weaker?

Low Basis Thm (Jockusch and Soare). T has a path P s.t. P ′ 6T ∅′.

Such a P is called low.

This theorem relativizes: If the binary tree T is computablerelative to X then T has a path P s.t. (P ⊕ X)′ 6T X ′.

Computable Entailment

Second-Order Statements

Versions of KL are second-order statements, involvingquantification over first-order (finite) objects and second-order(countably infinite) objects.

We can encode finite objects as natural numbers: e.g., strings,rationals, finite sets, . . .

We can encode countably infinite objects as sets of naturalnumbers: e.g., infinite sequences, trees, groups, reals, . . .

So we might encode a σ ∈ 2<ω of length n as2σ(0) + 4σ(1) + · · ·+ 2nσ(n− 1).

Then a tree is just a particular kind of subset of N.

Thus we can work in second-order arithmetic.

Π12 Statements

Statements involving only first-order quantification are calledarithmetic.

Version of KL are of the form

∀X [Θ(X) → ∃Y Ψ(X ,Y )],

where Θ and Ψ are arithmetic.

We can think of such a statement as a problem:An instance is an X s.t. Θ(X) holds.A solution to X is a Y s.t. Ψ(X ,Y ) holds.

Solving an instance of WKL takes less power than solving aninstance of KL.

But what about multiple instances?

Π12 Statements

∀X [Θ(X) → ∃Y Ψ(X ,Y )],

Π12 Statements

∀X [Θ(X) → ∃Y Ψ(X ,Y )],

Π12 Statements

∀X [Θ(X) → ∃Y Ψ(X ,Y )],

Π12 Statements

∀X [Θ(X) → ∃Y Ψ(X ,Y )],

Turing Ideals

A Turing ideal is an I ⊆ 2N s.t. if B1, . . . ,Bn ∈ I and A iscomputable relative to B1, . . . ,Bn then A ∈ I.

A problem P holds in I if for every instance X of P in I, there is asolution Y to X in I.

P computably entails Q, written as P �c Q, if Q holds in everyTuring ideal in which P holds.

P and Q are computably equivalent if they hold in the sameTuring ideals.

A statement Φ of second-order arithmetic holds in I if Φ is truewhen ∃X and ∀X are replaced by ∃X ∈ I and ∀X ∈ I.

Turing Ideals

Building Turing Ideals

Clearly KL �c WKL and WKL �c WWKL.

BKL holds in every Turing ideal, so WWKL �c BKL.

The computable sets form a Turing ideal I, and WWKL does nothold in I, so BKL 2c WWKL.

Thm (Scott/Jockusch and Soare/Friedman). WKL 2c KL.

The proof uses the relativized Low Basis Theorem: If the binarytree T is computable relative to X then T has a path P s.t.(P ⊕ X)′ 6T X ′.

Thm (Yu and Simpson). WWKL 2c WKL.

The proof uses the theory of algorithmic randomness.

Computable equivalence

BKL, WWKL, WKL, and KL represent important complexity levels.

Computably equivalent to BKL (i.e., computably true):I the existence of algebraic closures of fieldsI Godel’s Completeness Theorem for theoriesI the Intermediate Value TheoremI the Tietze Extension Theorem for complete separable metric

spaces...

Computably equivalent to WWKL:I the Vitali Covering TheoremI the monotone convergence theorem for Lebesgue measure

on [0, 1]

I the existence of (relatively) Martin-Lof random sequences...

spaces...

on [0, 1]

spaces...

on [0, 1]

Computably equivalent to WKL:I the uniqueness of algebraic closures for fieldsI the existence of prime ideals for commutative ringsI the Compactness Theorem for first-order logicI the Extreme Value TheoremI Brouwer’s Fixed Point Theorem

Computably equivalent to KL:I the existence of maximal ideals for commutative ringsI the existence of bases for vector spacesI the Bolzano-Weierstraß TheoremI the existence of the Turing jump

Computably equivalent to WKL:I the uniqueness of algebraic closures for fieldsI the existence of prime ideals for commutative ringsI the Compactness Theorem for first-order logicI the Extreme Value TheoremI Brouwer’s Fixed Point Theorem

Computably equivalent to KL:I the existence of maximal ideals for commutative ringsI the existence of bases for vector spacesI the Bolzano-Weierstraß TheoremI the existence of the Turing jump

Part III: Reverse Mathematics

Stephen G. Simpson, Subsystems of Second Order Arithmetic

Second-Order Arithmetic and RCA0

Second-Order Arithmetic

We work in a language with number variables, set variables, andsymbols 0, 1, S, <,+, ·,∈.

Again we encode finite objects as natural numbers and infiniteobjects as sets of natural numbers.

Reverse Mathematics: fix a weak base system and calibrate thestrength of principles by considering implications over this system.

Often in terms of a few subsystems of second-order arithmetic.

Full second-order arithmetic consists of

I axioms for a discrete ordered commutative semiring

I comprehension:

∃X ∀n [n ∈ X ↔ ϕ(n)]

for all formulas ϕ s.t. X is not free in ϕ

I induction:

(ϕ(0) ∧ ∀n [ϕ(n) → ϕ(n + 1)] → ∀n ϕ(n)

for all formulas ϕ

We obtain subsystems by limiting comprehension and induction.

Full second-order arithmetic consists of

I axioms for a discrete ordered commutative semiring

I comprehension:

∃X ∀n [n ∈ X ↔ ϕ(n)]

for all formulas ϕ s.t. X is not free in ϕ

I induction:

(ϕ(0) ∧ ∀n [ϕ(n) → ϕ(n + 1)] → ∀n ϕ(n)

for all formulas ϕ

We obtain subsystems by limiting comprehension and induction.

A Hierarchy of Arithmetic Formulas

A bounded quantifier is one of the form ∀x < t or ∃x < t .

A bounded-quantifier formula is an arithmetic formula in whichall quantifiers are bounded.

A Σ0n formula is one of the form

∃x1 ∀x2 ∃x3 ∀x4 · · ·Qxn ϕ,

where ϕ is a bounded-quantifier formula and Q is ∃ if n is oddand ∀ if n is even.

A Π0n formula is one of the form

∀x1 ∃x2 ∀x3 ∃x4 · · ·Qxn ϕ,

where ϕ is a bounded-quantifier formula and Q is ∀ if n is oddand ∃ if n is even.

These formulas can have free variables.

∃x1 ∀x2 ∃x3 ∀x4 · · ·Qxn ϕ,

∀x1 ∃x2 ∀x3 ∃x4 · · ·Qxn ϕ,

∃x1 ∀x2 ∃x3 ∀x4 · · ·Qxn ϕ,

∀x1 ∃x2 ∀x3 ∃x4 · · ·Qxn ϕ,

∃x1 ∀x2 ∃x3 ∀x4 · · ·Qxn ϕ,

∀x1 ∃x2 ∀x3 ∃x4 · · ·Qxn ϕ,

The Weak Base System RCA0

RCA0 is obtained by restricting:

I comprehension to ∆01-comprehension:

∀n [ϕ(n) ↔ ψ(n)] → ∃X ∀n [n ∈ X ↔ ϕ(n)]

for all ϕ,ψ s.t. ϕ is Σ01 and ψ is Π0

1, and X is not free in ϕ

I induction to Σ01-induction:

(ϕ(0) ∧ ∀n [ϕ(n) → ϕ(n + 1)]) → ∀n ϕ(n)

for all Σ01 formulas ϕ

This choice of base system creates a tight connection betweenthis approach and computable entailment.

The Weak Base System RCA0

RCA0 is obtained by restricting:

∀n [ϕ(n) ↔ ψ(n)] → ∃X ∀n [n ∈ X ↔ ϕ(n)]

(ϕ(0) ∧ ∀n [ϕ(n) → ϕ(n + 1)]) → ∀n ϕ(n)

This choice of base system creates a tight connection betweenthis approach and computable entailment.

Some Equivalences over RCA0

Provable in RCA0

I the existence of algebraic closures of fieldsI Godel’s Completeness Theorem for theoriesI the Intermediate Value TheoremI the Tietze Extension Theorem for complete separable metric

spaces...

Provably equivalent to WWKL over RCA0:I the Vitali Covering TheoremI the monotone convergence theorem for Lebesgue measure

on [0, 1]

Provable in RCA0

I the existence of algebraic closures of fieldsI Godel’s Completeness Theorem for theoriesI the Intermediate Value TheoremI the Tietze Extension Theorem for complete separable metric

spaces...

Provably equivalent to WWKL over RCA0:I the Vitali Covering TheoremI the monotone convergence theorem for Lebesgue measure

on [0, 1]

Provably equivalent to WKL over RCA0:I the uniqueness of algebraic closures for fieldsI the existence of prime ideals for commutative ringsI the Compactness Theorem for first-order logicI the Extreme Value TheoremI Brouwer’s Fixed Point Theorem

Provably equivalent to KL over RCA0:I the existence of maximal ideals for commutative ringsI the existence of bases for vector spacesI the Bolzano-Weierstraß TheoremI the existence of the Turing jump

Computability and Definability

The Arithmetic Hierarchy

A first-order formula is one with no set variables.

A ⊆ N is defined in N by a first-order formula ϕ(y) if: k ∈ A iff ϕ(k)holds in N.

A set is Σ0n if it is defined in N by some Σ0

n first-order formula.

A set is Π0n if it is defined in N by some Π0

A set is ∆0n if it is both Σ0

n and Π0n.

A set is arithmetic if it is in one of these classes.

A first-order formula is one with no set variables.

A ⊆ N is defined in N by a first-order formula ϕ(y) if: k ∈ A iff ϕ(k)holds in N.

A set is Σ0n if it is defined in N by some Σ0

A set is Π0n if it is defined in N by some Π0

A set is ∆0n if it is both Σ0

n and Π0n.

A set is arithmetic if it is in one of these classes.

∆01 ∆0

2 ∆03 . . .

Σ01 Σ0

Π01 Π0

( (( ( (( (

Thm (Kleene). A is Σ01 iff A is c.e. Thus A is ∆0

1 iff A is computable.

Recall that Z ′ is the Halting Problem relative to Z .

Define X (n) as follows: X (0) = X and X (n+1) = (X (n))′.

Thm (Post). A set is Σ0n+1 iff it is c.e. relative to ∅(n), and is ∆0

n+1 iff itis computable relative to ∅(n).

∆01 ∆0

2 ∆03 . . .

Σ01 Σ0

Π01 Π0

( (( ( (( (

∆01 ∆0

2 ∆03 . . .

Σ01 Σ0

Π01 Π0

( (( ( (( (

∆01 ∆0

2 ∆03 . . .

Σ01 Σ0

Π01 Π0

( (( ( (( (

Relativizing the Arithmetic Hierarchy

All of this can be relativized to any S ⊆ N:

Consider arithmetic formulas with one free set variable X .

A ⊆ N is defined in (N, S) by ϕ(y ,X) if: k ∈ A iff ϕ(k , S) holds in N.

A set is Σ0n relative to S if it is defined in (N, S) by some Σ0

n formula.

A set is Π0n relative to S if it is defined in (N, S) by some Π0

n formula.

A set is ∆0n relative to S if it is both Σ0

n and Π0n relative to S.

Post’s Theorem holds in relativized form.

In particular, A is ∆01 relative to S iff A is computable relative to S.

n formula.

RCA0 and Computability

Recall that RCA0 is obtained by restricting:

∀n [ϕ(n) ↔ ψ(n)] → ∃X ∀n [n ∈ X ↔ ϕ(n)]

(ϕ(0) ∧ ∀n [ϕ(n) → ϕ(n + 1)]) → ∀n ϕ(n)

∆01-comprehension is (relative) computable comprehension.

Indeed, RCA stands for Recursive Comprehension Axiom.

RCA0 and Computability

Recall that RCA0 is obtained by restricting:

∀n [ϕ(n) ↔ ψ(n)] → ∃X ∀n [n ∈ X ↔ ϕ(n)]

(ϕ(0) ∧ ∀n [ϕ(n) → ϕ(n + 1)]) → ∀n ϕ(n)

∆01-comprehension is (relative) computable comprehension.

Indeed, RCA stands for Recursive Comprehension Axiom.

Models of RCA0

A model in the language of second-order arithmetic consists ofa first-order part N = (N; 0N , 1N , SN , <N ,+N , ·N) and asecond-order part S ⊆ 2N .

If N is the standard natural numbers, we call this an ω-modeland identify it with S.

Thm (Friedman). S is an ω-model of RCA0 iff S is a Turing ideal.

Cor. If RCA0 + P ` Q then P �c Q.

The converse does not always hold because non-ω-models ofRCA0 exist, but it often does.

Models of RCA0

The Reverse-Mathematical Universe

Mathematics in RCA0

Several theorems can be proved in RCA0, e.g. many basicproperties of the natural numbers and the reals, as well asI the existence of algebraic closures of fieldsI Godel’s Completeness Theorem for theoriesI the Intermediate Value TheoremI the Tietze Extension Theorem for complete separable metric

spaces...

But the computable sets form an ω-model of RCA0, so theoremsthat are not computably true cannot be proved in RCA0.

Limited induction also plays a role.

Thm (Yokoyama). BKL is not provable in RCA0.

Mathematics in RCA0

spaces...

Mathematics in RCA0

spaces...

Mathematics in RCA0

spaces...

Other Subsystems of Second-Order Arithmetic

ACA0: RCA0 + arithmetic comprehension:

∃X ∀n [n ∈ X ↔ ϕ(n)]

for all arithmetic ϕ s.t. X is not free in ϕ

ACA0 implies arithmetic induction.

A Turing ideal is an ω-model of ACA0 iff it is closed under jumps.

RCA0 + Σ01-comprehension implies ACA0.

KL is equivalent to ACA0 over RCA0. So are

I the existence of maximal ideals for commutative ringsI the existence of bases for vector spacesI the Bolzano-Weierstraß TheoremI the existence of the Turing jump

∃X ∀n [n ∈ X ↔ ϕ(n)]

KL is equivalent to ACA0 over RCA0.

So are

∃X ∀n [n ∈ X ↔ ϕ(n)]

WKL0: RCA0 + Weak Konig’s Lemma

WKL0 is arithmetically conservative over RCA0, i.e., if WKL0 provesan arithmetic statement, then so does RCA0.

So WKL0 has the same amount of induction as RCA0.

ω-models of WKL0 are also known as Scott sets.

Equivalents of WKL0

I the uniqueness of algebraic closures for fieldsI the existence of prime ideals for commutative ringsI the Compactness Theorem for first-order logicI the Extreme Value TheoremI Brouwer’s Fixed Point Theorem

Equivalents of WKL0

WWKL0: RCA0 + Weak Weak Konig’s Lemma

Equivalents of WWKL0:I the Vitali Covering TheoremI the monotone convergence theorem for Lebesgue measure

on [0, 1]

ATR0: RCA0 + arithmetic transfinite recursion

Equivalents of ATR0

I comparability of well-orderingsI Ulm’s Theorem on Abelian p-groupsI the Perfect Set Theorem...

Π11-CA0: RCA0 + Π1

1-comprehension

A Π11 formula is one of the form ∀X ϕ, where ϕ is arithmetic.

Equivalents of Π11-CA0

I every countable Abelian group is the direct sum of adivisible group and a reduced group

I the Cantor-Bendixson Theorem...

Equivalents of ATR0

Π11-CA0: RCA0 + Π1

1-comprehension

Equivalents of ATR0

Π11-CA0: RCA0 + Π1

1-comprehension

Equivalents of ATR0

Π11-CA0: RCA0 + Π1

1-comprehension

Relationships between Subsystems of Second-Order Arithmetic

Π11-CA0

↓ATR0

↓ACA0

↓WKL0

↓WWKL0

↓RCA0

Ramsey’s Theorem

[X ]n is the set of n-element subsets of X .

A k-coloring of [X ]n is a map c : [X ]n → k .

A set H ⊆ X is homogeneous for c if |c([H]n)| = 1.

Ramsey’s Theorem for n-tuples and k colors (RTnk ): Every

k-coloring of [N]n has an infinite homogeneous set.

For j, k > 2, RTnj and RTn

k are equivalent over RCA0.

RTn<∞ is ∀k RTn

k and RT is ∀n ∀k RTnk .

Ramsey’s Theorem

RTn<∞ is ∀k RTn

Ramsey’s Theorem

RTn<∞ is ∀k RTn

The Reverse Mathematics of Ramsey’s Theorem

Thm (Jockusch/Simpson). For n > 3, RCA0 ` RTn2 ↔ ACA0.

Thm (Seetapun). RCA0 + RT22 0 ACA0.

Thm (Hirst). WKL0 0 RT22.

Thm (Liu). RCA0 + RT22 0 WWKL0.

RCA0 ` RT1k for any k ∈ N, but:

Thm (Hirst). RCA0 0 RT1<∞.

Thm (Jockusch). ACA0 0 RT.

Some consequences of RT22

Ascending / Descending Sequence Principle (ADS): Every infinitelinear order has an infinite ascending or descending sequence.

Chain / Antichain Principle (CAC): Every infinite partial order hasan infinite chain or antichain.

Atomic Model Theorem (AMT): Every complete atomic theoryhas an atomic model.

Existence of Diagonally Nonrecursive Functions (DNR): For everyX , there is a function f s.t. f (e) 6= ΦX

e(e) for all e.

A Small Part of the Universe Between RCA0 and ACA0

��

��CAC

��

��AMT DNR

Combined results of Yu and Simpson; Giusto and Simpson;Ambos-Spies, Kjos-Hanssen, Lempp, and Slaman; Hirschfeldt andShore; Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman;Hirschfeldt, Shore, and Slaman; Liu; and Lerman, Solomon, andTowsner.

A Larger Part of the Universe Between RCA0 and ACA0

BSig2+Pi01G

POS+WWKL

FS3 RT22+WKL+RRT32

SRT22 ASRAM

ISig2+AMT

CAC+WKL

BSig2+RRT22 POS1 RAN2 DTCp POS2 BSig2+RAN2

RAN1 WWKL

RWKL ASRT22 SPT22 SEM+SADS D22 SIPT22

ADS SCAC+CCAC

StCOH SADS+CADS CCAC

PART RT12

RCOLOR2

BSig2+COH+RRT22

StCRT22 StCADS BSig2+COH CRT22+BSig2 BSig2+CADS

COH+WKL

RWWKL1 RRT22

RWKL1 EM+ADS SRT22+COH PT22

EM+BSig2

STS2 RCOLOR3

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