Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard counterparts of several weak axioms . Keita Yokoyama (joint work with Kojiro Higuchi)

.

Mathematical Institute, Tohoku University

July 23, 2010

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Outline 1

Introduction Related topics Nonstandard second-order arithmetic Axioms for nonstandard arithmetic

2

Weak nonstandard axioms from recursion A nonstandard approach for recursion MLR∗ and DNR∗ Nonstandard extension

3

Conservativity r-extension and d-extension Proof of conservation results

4

Future work Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Outline 1

Introduction Related topics Nonstandard second-order arithmetic Axioms for nonstandard arithmetic

2

Weak nonstandard axioms from recursion A nonstandard approach for recursion MLR∗ and DNR∗ Nonstandard extension

3

Conservativity r-extension and d-extension Proof of conservation results

4

Future work Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard analysis and arithmetic (Model theoretic) nonstandard arguments for reverse mathematics in WKL0 and ACA0 (Tanaka, Yamzaki, Sakamoto, Y). Comparing axioms of nonstandard arithmetic and second-order arithmetic (Keisler, Henson, Kaufmann,. . . ). Formalizing analysis and nonstandard analysis within nonstandard arithmetic, and doing Reverse Mathematics (Impens, Sanders, Y). ⇐ Motivated by Prof. Fuchino’s question. Searching nonstandard counterparts of systems of second-order arithmetic (Keisler, Y). Comparing axioms of nonstandard arithmetic and weak axioms of arithmetic (Impens, Sanders). Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard analysis and arithmetic (Model theoretic) nonstandard arguments for reverse mathematics in WKL0 and ACA0 (Tanaka, Yamzaki, Sakamoto, Y). Comparing axioms of nonstandard arithmetic and second-order arithmetic (Keisler, Henson, Kaufmann,. . . ). Formalizing analysis and nonstandard analysis within nonstandard arithmetic, and doing Reverse Mathematics (Impens, Sanders, Y). ⇐ Motivated by Prof. Fuchino’s question. Searching nonstandard counterparts of systems of second-order arithmetic (Keisler, Y). Comparing axioms of nonstandard arithmetic and weak axioms of arithmetic (Impens, Sanders). Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard analysis and arithmetic (Model theoretic) nonstandard arguments for reverse mathematics in WKL0 and ACA0 (Tanaka, Yamzaki, Sakamoto, Y). Comparing axioms of nonstandard arithmetic and second-order arithmetic (Keisler, Henson, Kaufmann,. . . ). Formalizing analysis and nonstandard analysis within nonstandard arithmetic, and doing Reverse Mathematics (Impens, Sanders, Y). ⇐ Motivated by Prof. Fuchino’s question. Searching nonstandard counterparts of systems of second-order arithmetic (Keisler, Y). Comparing axioms of nonstandard arithmetic and weak axioms of arithmetic (Impens, Sanders). Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard analysis and arithmetic (Model theoretic) nonstandard arguments for reverse mathematics in WKL0 and ACA0 (Tanaka, Yamzaki, Sakamoto, Y). Comparing axioms of nonstandard arithmetic and second-order arithmetic (Keisler, Henson, Kaufmann,. . . ). Formalizing analysis and nonstandard analysis within nonstandard arithmetic, and doing Reverse Mathematics (Impens, Sanders, Y). ⇐ Motivated by Prof. Fuchino’s question. Searching nonstandard counterparts of systems of second-order arithmetic (Keisler, Y). Comparing axioms of nonstandard arithmetic and weak axioms of arithmetic (Impens, Sanders). Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard analysis and arithmetic (Model theoretic) nonstandard arguments for reverse mathematics in WKL0 and ACA0 (Tanaka, Yamzaki, Sakamoto, Y). Comparing axioms of nonstandard arithmetic and second-order arithmetic (Keisler, Henson, Kaufmann,. . . ). Formalizing analysis and nonstandard analysis within nonstandard arithmetic, and doing Reverse Mathematics (Impens, Sanders, Y). ⇐ Motivated by Prof. Fuchino’s question. Searching nonstandard counterparts of systems of second-order arithmetic (Keisler, Y). Comparing axioms of nonstandard arithmetic and weak axioms of arithmetic (Impens, Sanders). Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard analysis and arithmetic (Model theoretic) nonstandard arguments for reverse mathematics in WKL0 and ACA0 (Tanaka, Yamzaki, Sakamoto, Y). Comparing axioms of nonstandard arithmetic and second-order arithmetic (Keisler, Henson, Kaufmann,. . . ). Formalizing analysis and nonstandard analysis within nonstandard arithmetic, and doing Reverse Mathematics (Impens, Sanders, Y). ⇐ Motivated by Prof. Fuchino’s question. Searching nonstandard counterparts of systems of second-order arithmetic (Keisler, Y). Comparing axioms of nonstandard arithmetic and weak axioms of arithmetic (Impens, Sanders). Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Language of nonstandard second order arithmetic Language of second-order arithmetic: L2 = {0, 1, =, +, ·, <, ∈}. We expand L2 to L∗2 . Definition (Language L∗2 ) Language of nonstandard second-order arithmetic (L∗2 ) consists of s number variables: x s , y s , . . ., ∗ number variables: x ∗ , y ∗ , . . ., s set variables: X s , Y s , . . ., ∗ set variables: X ∗ , Y ∗ , . . ., s symbols: 0s , 1s , =s , +s , ·s ,
Keita Yokoyama

.

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

s-structure and ∗-structure

M s : range of x s , y s , . . ., M ∗ : range of x ∗ , y ∗ , . . ., S s : range of X s , Y s , . . ., S ∗ : range of X ∗ , Y ∗ , . . .

N, the standard numbers N∗ , the nonstandard numbers subsets of N subsets of N∗ .

V s = (M s , S s ; 0s , 1s , . . . ): s-L2 structure. V ∗ = (M ∗ , S ∗ ; 0∗ , 1∗ , . . . ): ∗-L2 structure. √ : M s ∪ S s → M ∗ ∪ S ∗ : embedding. We usually regard M s is a subset of M ∗ .

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Notations: Let φ be an L2 -formula.

φs : L∗2 formula constructed by adding s to L2 symbols in φ.

φ∗ : L∗2 formula constructed by adding ∗ to L2 symbols in φ. √ xˇs := (x s ). √ Xˇs := (X s ). We usually omit s and ∗ of relations =, ≤, ∈. We often say “φ holds in V s (in V ∗ )” when φs (φ∗ ) holds.

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Systems of nonstandard arithmetic Definition (base system) RCA0 ns consists of the following:

(RCA0 )s . √ : V s → V ∗ is an injective homomorphism. M ∗ is an end extension of M s . finite standard part principle: ∀X ∗ (card(X ∗ ) ∈ M s → ∃Y s ∀x s (x s ∈ Y s ↔ xˇs ∈ X ∗ ).

Σ01 saturation: for any φ ∈ Σ01 , (∀x s ∃y ∗ φ(xˇs , y ∗ )∗ → ∃y ∗ ∀x s φ(xˇs , y ∗ )∗ ). Σ00 transfer principle: for any φ ∈ Σ00 , ∀x s ∀X s (φ(x s , X s )s ↔ φ(xˇs , Xˇs )∗ ). Σ11 equivalence: for any sentence φ ∈ Σ11 , (φs ↔ φ∗ ). Keita Yokoyama

.

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Systems of nonstandard arithmetic In short, Definition (base system) RCA0 ns consists of the following:

(RCA0 )s : base theory of Reverse Mathematics plus innocent nonstandard axioms. RCA0 ns is a weak nonstandard theory that can still define.real numbers, continuous functions, . . . . Theorem RCA0 ns is a conservative extension of RCA0 . . Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Systems of nonstandard arithmetic In short, Definition (base system) RCA0 ns consists of the following:

(RCA0 )s : base theory of Reverse Mathematics plus innocent nonstandard axioms. RCA0 ns is a weak nonstandard theory that can still define.real numbers, continuous functions, . . . . Theorem RCA0 ns is a conservative extension of RCA0 . . Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Systems of nonstandard arithmetic In short, Definition (base system) RCA0 ns consists of the following:

(RCA0 )s : base theory of Reverse Mathematics plus innocent nonstandard axioms. RCA0 ns is a weak nonstandard theory that can still define.real numbers, continuous functions, . . . . Theorem RCA0 ns is a conservative extension of RCA0 . . Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Systems of nonstandard arithmetic

Definition WKL0 ns consists of RCA0 ns plus the standard part principle (STP) which asserts that

STP : ∀X ∗ ∃Y s ∀x s (x s ∈ Y s ↔ xˇs ∈ X ∗ ). Theorem

.

WKL0 ns is a conservative extension of WKL0 .

. Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Systems of nonstandard arithmetic

Definition ACA0 ns consists of WKL0 ns plus Σ11 transfer principle (Σ11 TP)

Σ11 TP : ∀x s ∀X s (φ(x s , X s )s ↔ φ(xˇs , Xˇs )∗ ) for any for any φ ∈ Σ11 . .

Theorem ACA0 ns is a conservative extension of ACA0 .

Keita Yokoyama

.

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Systems of nonstandard arithmetic Definition WWKL0 ns consists of RCA0 ns plus the Loeb measure principle (LMP) which asserts that ∗

LMP : ∀H ∗ ∈ N∗ \ Ns ∀T ∗ ⊆ 20 ∗ 2H → ∃σ∗ ∈ T ∗ lh(σ∗ ) = H ∗ ∧ σ∗ ∩ Ns ∈ V s . . LMP :an NS-tree which has a positive measure has a standard path. Theorem (Simpson, Y) WWKL0 ns is a conservative extension of WWKL0 .

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard reverse mathematics Within RCA0 ns , we can develop basic part of nonstandard analysis. Moreover, within RCA0 ns , “Lebesgue measure is the standard part of Loeb measure” ⇔ WWKL0 ns . “Riemann integral can be infinitesimally approximated by hyperfinite Riemann sum” ⇔ WWKL0 ns . Every continuous function can be infinitesimally approximated by hyperfinite broken line. ⇔ WKL0 ns . Every compact separable metric space is a standard part of a nonstandard metric space whose points are all standard. ⇔ WKL0 ns . Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard reverse mathematics “If {ans } is a bounded sequence of real numbers, then for some nonstandard ω, st(aω∗ ) is an accumulation value of {ans }” (nonstandard Bolzano-Weierstraß theorem) ⇐ ACA0 ns . “If {fn } is a normal family of holomorphic functions on a bounded domain, then for some nonstandard ω, st(fω∗ ) is an accumulation value of {fns }” ⇐ ACA0 ns . These Reverse Mathematics phenomena show that WWKL0 ns , WKL0 ns , . . . are nice nonstandard counterparts of second-order arithmetic. ⇒ Can we find nonstandard counterparts of other systems such as DNR, MLR, COH,. . . ? Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard reverse mathematics “If {ans } is a bounded sequence of real numbers, then for some nonstandard ω, st(aω∗ ) is an accumulation value of {ans }” (nonstandard Bolzano-Weierstraß theorem) ⇐ ACA0 ns . “If {fn } is a normal family of holomorphic functions on a bounded domain, then for some nonstandard ω, st(fω∗ ) is an accumulation value of {fns }” ⇐ ACA0 ns . These Reverse Mathematics phenomena show that WWKL0 ns , WKL0 ns , . . . are nice nonstandard counterparts of second-order arithmetic. ⇒ Can we find nonstandard counterparts of other systems such as DNR, MLR, COH,. . . ? Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard reverse mathematics “If {ans } is a bounded sequence of real numbers, then for some nonstandard ω, st(aω∗ ) is an accumulation value of {ans }” (nonstandard Bolzano-Weierstraß theorem) ⇐ ACA0 ns . “If {fn } is a normal family of holomorphic functions on a bounded domain, then for some nonstandard ω, st(fω∗ ) is an accumulation value of {fns }” ⇐ ACA0 ns . These Reverse Mathematics phenomena show that WWKL0 ns , WKL0 ns , . . . are nice nonstandard counterparts of second-order arithmetic. ⇒ Can we find nonstandard counterparts of other systems such as DNR, MLR, COH,. . . ? Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Outline 1

Introduction Related topics Nonstandard second-order arithmetic Axioms for nonstandard arithmetic

2

Weak nonstandard axioms from recursion A nonstandard approach for recursion MLR∗ and DNR∗ Nonstandard extension

3

Conservativity r-extension and d-extension Proof of conservation results

4

Future work Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard Turing Machine Can we expand recursive notion into nonstandard analysis? Let {e }As (x ) be the result of the calculation of e-th Turing machine with an input x and an oracle A by s steps. For a (standard or nonstandard) natural number α, code(α) denotes the finite set coded by α. If α is nonstandard, code(α) may be a hyperfinite set. Definition For a nonstandard natural number α s {e }α (x ) = y ⇔ ∃s ∈ Ns {e }α s (x ) = y

where α  s = code(α)  s. By this definition, we can use the notion α-recursive for a nonstandard natural number α. Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard Turing Machine Can we expand recursive notion into nonstandard analysis? Let {e }As (x ) be the result of the calculation of e-th Turing machine with an input x and an oracle A by s steps. For a (standard or nonstandard) natural number α, code(α) denotes the finite set coded by α. If α is nonstandard, code(α) may be a hyperfinite set. Definition For a nonstandard natural number α s {e }α (x ) = y ⇔ ∃s ∈ Ns {e }α s (x ) = y

where α  s = code(α)  s. By this definition, we can use the notion α-recursive for a nonstandard natural number α. Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard Turing Machine Can we expand recursive notion into nonstandard analysis? Let {e }As (x ) be the result of the calculation of e-th Turing machine with an input x and an oracle A by s steps. For a (standard or nonstandard) natural number α, code(α) denotes the finite set coded by α. If α is nonstandard, code(α) may be a hyperfinite set. Definition For a nonstandard natural number α s {e }α (x ) = y ⇔ ∃s ∈ Ns {e }α s (x ) = y

where α  s = code(α)  s. By this definition, we can use the notion α-recursive for a nonstandard natural number α. Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

MLR∗ and DNR∗ Using the notion of α-recursive for a nonstandard natural number α, we define nonstandard axioms for some recursive notions. ¨ MLR: for any set A , there exists X such that X is Martin-Lof random relative to A .

MLR∗ : for any nonstandard number α, there exists X such ¨ random relative to α. that X is Martin-Lof DNR: for any set A , there exists X such that X is diagonally non-recursive relative to A . DNR∗ : for any nonstandard number α, there exists X such that X is diagonally non-recursive relative to α.

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

MLR∗ and DNR∗ Using the notion of α-recursive for a nonstandard natural number α, we define nonstandard axioms for some recursive notions. ¨ MLR: for any set A , there exists X such that X is Martin-Lof random relative to A .

MLR∗ : for any nonstandard number α, there exists X such ¨ random relative to α. that X is Martin-Lof DNR: for any set A , there exists X such that X is diagonally non-recursive relative to A . DNR∗ : for any nonstandard number α, there exists X such that X is diagonally non-recursive relative to α.

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

MLR∗ and DNR∗ Using the notion of α-recursive for a nonstandard natural number α, we define nonstandard axioms for some recursive notions. ¨ MLR: for any set A , there exists X such that X is Martin-Lof random relative to A .

MLR∗ : for any nonstandard number α, there exists X such ¨ random relative to α. that X is Martin-Lof DNR: for any set A , there exists X such that X is diagonally non-recursive relative to A . DNR∗ : for any nonstandard number α, there exists X such that X is diagonally non-recursive relative to α.

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Main theorems Is DNR∗ (MLR∗ ) a nonstandard counterpart of DNR (MLR)? ⇒ Yes. Theorem

DNR∗ is a nonstandard counterpart of DNR. Theorem (Simpson, Y)

MLR∗ is a nonstandard counterpart of MLR.

.

Here, “nonstandard counterpart” means that it is a ‘nonstandard . extension’ and a ‘conservative extension (w.r.t. standard formulas)’.

Keita Yokoyama

.

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Main theorems Is DNR∗ (MLR∗ ) a nonstandard counterpart of DNR (MLR)? ⇒ Yes. Theorem

DNR∗ is a nonstandard counterpart of DNR. Theorem (Simpson, Y)

MLR∗ is a nonstandard counterpart of MLR.

.

Here, “nonstandard counterpart” means that it is a ‘nonstandard . extension’ and a ‘conservative extension (w.r.t. standard formulas)’.

Keita Yokoyama

.

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Main theorems Is DNR∗ (MLR∗ ) a nonstandard counterpart of DNR (MLR)? ⇒ Yes. Theorem

DNR∗ is a nonstandard counterpart of DNR. Theorem (Simpson, Y)

MLR∗ is a nonstandard counterpart of MLR.

.

Here, “nonstandard counterpart” means that it is a ‘nonstandard . extension’ and a ‘conservative extension (w.r.t. standard formulas)’.

Keita Yokoyama

.

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

WWKL0 ns and MLR∗

Note that Fact

. .

1

¨ MLR is equivalent to weak weak Konig’s lemma over RCA0 .

2

MLR∗ is equivalent to LMP over RCA0 ns .

Thus, ‘MLR∗ is a nonstandard counterpart of MLR’ . is just a restatement of ‘WWKL0 ns = RCA0 ns + LMP is a nonstandard counterpart of WWKL0 ’. .

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard extension Within RCA0 ns , we can code subsets of N with nonstandard numbers. Lemma For any set A , there exists α ∈ N∗ such that

∀n ∈ Ns n ∈ A ↔ n ∈ code(α). Thus, we have the following:

.

Proposition RCA0 ns + DNR∗ proves DNR. . Proposition RCA0 ns + MLR∗ proves MLR. Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

MLR∗ and MLR Is MLR∗ strictly stronger than MLR? ⇒ Yes. Fact RCA0 ns + MLR∗ + Σ01 TP proves that there exists a 2-random real. RCA0 ns + MLR + Σ01 TP does not prove that there exists a 2-random real. Thus, .

Proposition RCA0 ns + MLR does not prove MLR∗ .

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

MLR∗ and MLR Is MLR∗ strictly stronger than MLR? ⇒ Yes. Fact RCA0 ns + MLR∗ + Σ01 TP proves that there exists a 2-random real. RCA0 ns + MLR + Σ01 TP does not prove that there exists a 2-random real. Thus, .

Proposition RCA0 ns + MLR does not prove MLR∗ .

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

MLR∗ and MLR Is MLR∗ strictly stronger than MLR? ⇒ Yes. Fact RCA0 ns + MLR∗ + Σ01 TP proves that there exists a 2-random real. RCA0 ns + MLR + Σ01 TP does not prove that there exists a 2-random real. Thus, .

Proposition RCA0 ns + MLR does not prove MLR∗ .

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

DNR∗ and DNR Is DNR∗ strictly stronger than DNR? ⇒ Yes. Fact RCA0 ns + DNR∗ + Σ01 TP proves that there exists a diagonally non-recursive set relative to 0′ . RCA0 ns + DNR + Σ01 TP does not prove that there exists a diagonally non-recursive set relative to 0′ . Thus, .

Proposition RCA0 ns + DNR does not prove DNR∗ .

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

DNR∗ and DNR Is DNR∗ strictly stronger than DNR? ⇒ Yes. Fact RCA0 ns + DNR∗ + Σ01 TP proves that there exists a diagonally non-recursive set relative to 0′ . RCA0 ns + DNR + Σ01 TP does not prove that there exists a diagonally non-recursive set relative to 0′ . Thus, .

Proposition RCA0 ns + DNR does not prove DNR∗ .

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

DNR∗ and DNR Is DNR∗ strictly stronger than DNR? ⇒ Yes. Fact RCA0 ns + DNR∗ + Σ01 TP proves that there exists a diagonally non-recursive set relative to 0′ . RCA0 ns + DNR + Σ01 TP does not prove that there exists a diagonally non-recursive set relative to 0′ . Thus, .

Proposition RCA0 ns + DNR does not prove DNR∗ .

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Outline 1

Introduction Related topics Nonstandard second-order arithmetic Axioms for nonstandard arithmetic

2

Weak nonstandard axioms from recursion A nonstandard approach for recursion MLR∗ and DNR∗ Nonstandard extension

3

Conservativity r-extension and d-extension Proof of conservation results

4

Future work Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Conservativity

Now, we prove the conservation results. Theorem (review) RCA0 ns + MLR∗ is a conservative extension of RCA0 + MLR. Theorem . . RCA0 ns + DNR∗ is a conservative extension of RCA0 + DNR We use the similar way to prove these theorems.

.

We first prepare ‘r-extension’ and ‘d-extension’.

. Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

randomness and r-extension For S ⊆ S ′ ⊆ P(N), S ⊆r S ′ ⇔ for any A ∈ S ′ there exists X ∈ S such that X is random relative to A . Proposition (Downey et. el. 2005, Reiman and Slaman 2010) ¨ random, then for any nonempty Π01 -class If X ∈ 2ω is Martin-Lof ¨ random P ⊆ 2ω there exists A ∈ P such that X is Martin-Lof relative to A . Combining Harrington’s forcing with the previous proposition, we have the following. . Lemma (r-extension) Let (M , S ) |= RCA0 + MLR be a countable model. Then, there ¯ ⊇r S such that (M , S ¯ ) |= WKL0 . exists an ω-extension S . Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

randomness and r-extension For S ⊆ S ′ ⊆ P(N), S ⊆r S ′ ⇔ for any A ∈ S ′ there exists X ∈ S such that X is random relative to A . Proposition (Downey et. el. 2005, Reiman and Slaman 2010) ¨ random, then for any nonempty Π01 -class If X ∈ 2ω is Martin-Lof ¨ random P ⊆ 2ω there exists A ∈ P such that X is Martin-Lof relative to A . Combining Harrington’s forcing with the previous proposition, we have the following. . Lemma (r-extension) Let (M , S ) |= RCA0 + MLR be a countable model. Then, there ¯ ⊇r S such that (M , S ¯ ) |= WKL0 . exists an ω-extension S . Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

DNR and Π01 -class

To prove conservativity for DNR∗ , we first prepare the following proposition for DNR-functions. Proposition If f is diagonally non-recursive, then for any nonempty Π01 -class P ⊆ 2ω there exists A ∈ P such that there exists g ≤T f which is diagonally non-recursive relative to A .

. Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

DNR and Π01 -class Sketch of the proof. For any σ ∈ 2<ω and for any T ⊆ 2<ω , we define T (σ) by T (σ) = {τ ∈ T | (∀e < lh(σ))[{e }τlh(τ) (e ) ↓=⇒ σ(e ) , {e }τlh(τ) (e )]}. Choose a recursive binary tree T such that [T ] = P and a recursive function q ∈ S such that

{q(s , σ)}(x ) := (least z )[(∃y )(∀τ ∈ T (σ) ∩ 2y )[z = {s }τlh(τ) (s ) ↓]]. ∩

Let g (s ) = f (q(s , g  s )) and Q = s [T (g  s )]. Then g ≤T f and Q , ∅. ∩ ( s [T (g  s )] , ∅ can be proved by induction on s.) Take A ∈ Q arbitrary, then g is diagonally non-recursive relative to .  A since τ ∈ T (g  e + 1) implies g (e ) , {e }τ (e ). Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

DNR and d-extension For S ⊆ S ′ ⊆ P(N), S ⊆d S ′ ⇔ for any A ∈ S ′ there exists g ∈ S such that g is diagonally non-recursive relative to A . Proposition (review) If f is diagonally non-recursive, then for any nonempty Π01 -class P ⊆ 2ω there exists A ∈ P such that there exists g ≤T f which is diagonally non-recursive relative to A . Combining Harrington’s forcing with the previous proposition, we have the following. . Lemma (d-extension) Let (M , S ) |= RCA0 + DNR be a countable model. Then, there ¯ ⊇d S such that (M , S ¯ ) |= WKL0 . exists an ω-extension S . Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

DNR and d-extension For S ⊆ S ′ ⊆ P(N), S ⊆d S ′ ⇔ for any A ∈ S ′ there exists g ∈ S such that g is diagonally non-recursive relative to A . Proposition (review) If f is diagonally non-recursive, then for any nonempty Π01 -class P ⊆ 2ω there exists A ∈ P such that there exists g ≤T f which is diagonally non-recursive relative to A . Combining Harrington’s forcing with the previous proposition, we have the following. . Lemma (d-extension) Let (M , S ) |= RCA0 + DNR be a countable model. Then, there ¯ ⊇d S such that (M , S ¯ ) |= WKL0 . exists an ω-extension S . Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

MLR∗ is conservative over MLR

By the r-extension lemma and Tanaka’s self-embedding theorem for WKL0 , any coutable model of RCA0 + MLR can be expanded into a model of RCA0 ns + MLR∗ as follows.  

(M , S )

⊆r

¯) (M , S



|= MLR

(*)

|= WKL0

(**)



η

¯  I) (I, S

(e

¯) (M , S



(*) r-extension (**) Tanaka’s self-embedding theorem

¯ ), η  (M , S ) is a model of RCA0 ns + MLR∗ . Then, (M , S ), (M , S Thus, RCA0 ns + MLR∗ is a conservative extension of RCA0 + MLR.

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

MLR∗ is conservative over MLR

By the r-extension lemma and Tanaka’s self-embedding theorem for WKL0 , any coutable model of RCA0 + MLR can be expanded into a model of RCA0 ns + MLR∗ as follows.  

(M , S )

⊆r

¯) (M , S



|= MLR

(*)

|= WKL0

(**)



η

¯  I) (I, S

(e

¯) (M , S



(*) r-extension (**) Tanaka’s self-embedding theorem

¯ ), η  (M , S ) is a model of RCA0 ns + MLR∗ . Then, (M , S ), (M , S Thus, RCA0 ns + MLR∗ is a conservative extension of RCA0 + MLR.

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

MLR∗ is conservative over MLR

By the r-extension lemma and Tanaka’s self-embedding theorem for WKL0 , any coutable model of RCA0 + MLR can be expanded into a model of RCA0 ns + MLR∗ as follows.  

(M , S )

⊆r

¯) (M , S



|= MLR

(*)

|= WKL0

(**)



η

¯  I) (I, S

(e

¯) (M , S



(*) r-extension (**) Tanaka’s self-embedding theorem

¯ ), η  (M , S ) is a model of RCA0 ns + MLR∗ . Then, (M , S ), (M , S Thus, RCA0 ns + MLR∗ is a conservative extension of RCA0 + MLR.

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

DNR∗ is conservative over DNR By the d-extension lemma and Tanaka’s self-embedding theorem for WKL0 , any coutable model of RCA0 + DNR can be expanded into a model of RCA0 ns + DNR∗ as follows.  

(M , S )

⊆d

¯) (M , S



|= DNR

(*)

|= WKL0

(**)



η

¯  I) (I, S

(e

¯) (M , S



(*) d-extension (**) Tanaka’s self-embedding theorem

¯ ), η  (M , S ) is a model of RCA0 ns + DNR∗ . Then, (M , S ), (M , S Thus, RCA0 ns + DNR∗ is a conservative extension of RCA0 + DNR.

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

DNR∗ is conservative over DNR By the d-extension lemma and Tanaka’s self-embedding theorem for WKL0 , any coutable model of RCA0 + DNR can be expanded into a model of RCA0 ns + DNR∗ as follows.  

(M , S )

⊆d

¯) (M , S



|= DNR

(*)

|= WKL0

(**)



η

¯  I) (I, S

(e

¯) (M , S



(*) d-extension (**) Tanaka’s self-embedding theorem

¯ ), η  (M , S ) is a model of RCA0 ns + DNR∗ . Then, (M , S ), (M , S Thus, RCA0 ns + DNR∗ is a conservative extension of RCA0 + DNR.

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

DNR∗ is conservative over DNR By the d-extension lemma and Tanaka’s self-embedding theorem for WKL0 , any coutable model of RCA0 + DNR can be expanded into a model of RCA0 ns + DNR∗ as follows.  

(M , S )

⊆d

¯) (M , S



|= DNR

(*)

|= WKL0

(**)



η

¯  I) (I, S

(e

¯) (M , S



(*) d-extension (**) Tanaka’s self-embedding theorem

¯ ), η  (M , S ) is a model of RCA0 ns + DNR∗ . Then, (M , S ), (M , S Thus, RCA0 ns + DNR∗ is a conservative extension of RCA0 + DNR.

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Nonstandard counterparts of second-order arithmetic

We have the following nonstandard counterparts of systems of second-order arithmetic. second-order arithmetic RCA0 RCA0 + DNR RCA0 + MLR = WWKL0 WKL0 ACA0

nonstandard second-order arithmetic RCA0 ns RCA0 ns + DNR∗ RCA0 ns + MLR∗ = WWKL0 ns ns RCA0 + STP = WKL0 ns RCA0 ns + STP + Σ11 TP = ACA0 ns

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Outline 1

Introduction Related topics Nonstandard second-order arithmetic Axioms for nonstandard arithmetic

2

Weak nonstandard axioms from recursion A nonstandard approach for recursion MLR∗ and DNR∗ Nonstandard extension

3

Conservativity r-extension and d-extension Proof of conservation results

4

Future work Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Questions

Since RCA0 ns + MLR∗ is equivalent to WWKL0 ns , there are some nice Reverse Mathematics phenomena for MLR∗ . Question Is there some nice Reverse Mathematics phenomena for DNR or DNR∗ ?

. Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Future work

Does the same argument work for other recursive notions? For example, does this argument work for the cohesive axiom?

COH: for any sequence ⟨An | n ∈ N⟩, there exists an infinite set X such that X is ⟨An ⟩ cohesive. COH∗ : for any nonstandard sequence ⟨αn | n < β⟩, there exists an infinite set X such that X is ⟨code(αn ) | n ∈ N⟩ cohesive.

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Future work

Question Is RCA0 ns + COH∗ a conservative extension of RCA0 + COH? If the following question is true, we may apply the previous. argument and answer this question. . Question Let X is a recursive cohesive set, and let P be a non-empty Π01 -class. Then, is there a set A ∈ P such that X is A -recursive cohesive?

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Future work Theorem (folklore?) IΣ1 + Σn TP is a conservative extension of BΣn+1 .

.

Theorem

. − RCA0 ns + WKL0 + Σ01 TP is a conservative extension of WKL0 +BΣ2 . Fact RCA0 ns + Σ01 TP+COH∗ ⊢ RT22 .

.

Can we use these nonstandard arguments to analyze Ramsey theory for pairs? . Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

Future work Theorem (folklore?) IΣ1 + Σn TP is a conservative extension of BΣn+1 .

.

Theorem

. − RCA0 ns + WKL0 + Σ01 TP is a conservative extension of WKL0 +BΣ2 . Fact RCA0 ns + Σ01 TP+COH∗ ⊢ RT22 .

.

Can we use these nonstandard arguments to analyze Ramsey theory for pairs? . Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

References. Chris Impens and Sam Sanders, Transfer and a supremum principle for ERNA. J. Symbolic Logic 73 (2):689–710, 2008. H. Jerome Keisler, Nonstandard arithmetic and reverse mathematics, The Bulletin of Symbolic Logic, 12(1):100–125, 2006. Sam Sanders, A copy of several Reverse Mathematics, Logic Colloquium 2010. Stephen Simpson and Y, A nonstandard counterpart of WWKL, preprint. Kazuyuki Tanaka, The self-embedding theorem of WKL0 and a non-standard method, Annals of Pure and Applied Logic 84:41–49, 1997. Y, Formalizing non-standard arguments in second order arithmetic, preprint.

Keita Yokoyama

Nonstandard counterparts of several weak axioms

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

It remains to show that T (g  s ) is infinite. Clearly, T (g  0) = T is infinite. Assume that T (g  s ) is infinite. By definition, T (g  s + 1) = {τ ∈ T (g  s ) | {s }τlh(τ) (s ) ↓⇒ B ′ (s ) , {s }τlh(τ) (s )}. Take y ∈ ω arbitrary. If

(∀τ ∈ T (g  s ) ∩ 2y [g (s ) = {s }τlh(τ) (s ) ↓], then (∀x ) {q(s , g  s )}(x ) = g (s ). But f ∈ DNR and, therefore, f (q(s , g  s )) , {q(s , g  s )}(q(s , g  s )) = g (s ) = f (q(s , g  s )). A contradiction. Hence for any y ∈ ω,

¬(∀τ ∈ T (g  s ) ∩ 2y )[g (s ) = {s }τlh(τ) (s ) ↓]. So T (g  s + 1) is infinite since T (g  s ) is infinite. Keita Yokoyama

Nonstandard counterparts of several weak axioms