Reverse mathematics and Peano categoricity
Reverse mathematics and Peano categoricity
复制标题
逆向数学和皮亚诺范畴
DOI：
10.1016/j.apal.2012.10.014
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发表时间：
2013
影响因子：
0.8
通讯作者：
Yokoyama
中科院分区：
文献类型：
--
作者：
Stephen G. Simpson and Keita;Yokoyama
我们研究了自然数系统是二阶范畴的几个定理的逆数学状态。我们的结果之一如下。定义一个系统为三元组A，i，f使得A是一个集合且i∈A且f：A→A。如果i∈X且a（a∈X ‡ f（a）∈X），则子集X A称为归纳的。如果A的唯一归纳子集是A本身，则系统A，i，f被称为归纳的。定义一个Peano系统是一个归纳系统，使得f是一对一的，i是f的值域。皮亚诺系统的标准例子是N，0，S，其中N={0，1，2，.，n，...}=自然数的集合，S：N→N由S（n）=n+1给出，对于所有n∈N。考虑所有Peano系统同构于N，0，S的陈述。我们证明了这一陈述在逻辑上等价于RCA 0上的WKL 0。从这个和类似的等价物中，我们得出了一些基本的/哲学的结论。
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 ∀a (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 i∉the 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 WKL0over RCA0⁎. From this and similar equivalences we draw some foundational/philosophical consequences.