TY - JOUR

T1 - Reverse mathematics and Peano categoricity

AU - Simpson, Stephen G.

AU - Yokoyama, Keita

N1 - Funding Information:
E-mail addresses: simpson@math.psu.edu (S.G. Simpson), yokoyama.k.ai@m.titech.ac.jp (K. Yokoyama). URL: http://www.math.psu.edu/simpson (S.G. Simpson). 1 Yokoyama’s research was supported by a Japan Society for the Promotion of Science postdoctoral fellowship for young scientists, and by a grant from the John Templeton Foundation. 2 A crucial role in the development of reverse mathematics was played by H. Friedman [7,8]. We thank the referee for strongly suggesting that we include this historical comment.

PY - 2013/3

Y1 - 2013/3

N2 - 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.

AB - 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.

KW - Foundations of mathematics

KW - Peano system

KW - Proof theory

KW - Reverse mathematics

KW - Second-order arithmetic

KW - Second-order logic

UR - http://www.scopus.com/inward/record.url?scp=84871097688&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871097688&partnerID=8YFLogxK

U2 - 10.1016/j.apal.2012.10.014

DO - 10.1016/j.apal.2012.10.014

M3 - Article

AN - SCOPUS:84871097688

SN - 0168-0072

VL - 164

SP - 284

EP - 293

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

IS - 3

ER -