# Finite sets and infinite sets in weak intuitionistic arithmetic

**Date**: Wednesday 18 October 2017, 16:00 – 17:00**Location**: Mathematics Level 8, MALL 1 & 2, School of Mathematics**Type**: Logic, Seminars, Pure Mathematics**Cost**: Free

#### Takako Nemoto, JAIST. Part of the logic seminar series.

**Takako Nemoto, ***JAIST*

We consider, for a set A of natural numbers, the following notions of finiteness:

FIN1: There are k and m_0,...,m_{k-1} such that A = {m_0,...,m_{k-1}};

FIN2: There is an upper bound for A;

FIN3: There is an m such that |B| < m for all subsets B of A;

FIN4: It is not the case that, for all x, there is y > x such that y is in A;

FIN5: It is not the case that, for all m, there is a subset B of A such that |B| = m,

and infiniteness

INF1: There are no k and m_0,...,m_{k-1} such that A = {m_0,...,m_{k-1}};

INF2: There is no upper bound for A;

INF3: There is no m such that |B| < m for all subsets B of A;

INF4: For all x, there is a y > x such that y is in A;

INF5: For all m, there is a subset B of A such that |B| = m.

We systematically compare them in the method of constructive reverse mathematics. We show that the equivalence among them can be characterized by various combinations of induction axioms and non-constructive principles, including the axiom called bounded comprehension.