# Reverse mathematics and Peano categoricity

Stephen G. Simpson, Keita Yokoyama

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)

## Abstract

We investigate the reverse-mathematical status of several theorems to the effect that the natural number system is second-order categorical. One of our results is as follows. Define a system to be a triple A, i, f such that A is a set and i∈A and f:A→A. A subset X⊆A is said to be inductive if i∈X and Aa(a∈X⇒f(a)∈X). The system A, i, f is said to be inductive if the only inductive subset of A is A itself. Define a Peano system to be an inductive system such that f is one-to-one and iεthe range of f. The standard example of a Peano system is N,0,S where N={0,1,2,. .,n,. .}=the set of natural numbers and S:N→N is given by S(n)=n+1 for all n∈N. Consider the statement that all Peano systems are isomorphic to N,0,S. We prove that this statement is logically equivalent to WKL0 over RCA0*. From this and similar equivalences we draw some foundational/philosophical consequences.

Original language English 284-293 10 Annals of Pure and Applied Logic 164 3 https://doi.org/10.1016/j.apal.2012.10.014 Published - 2013 Mar Yes

## Keywords

• Foundations of mathematics
• Peano system
• Proof theory
• Reverse mathematics
• Second-order arithmetic
• Second-order logic

• Logic

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