Nonstandard counterparts of several weak axioms - Google Sites

Introduction

Weak nonstandard axioms from recursion

Conservativity

Future work

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

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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, =, +, ·,