Results 1 
1 of
1
FREE SETS AND REVERSE MATHEMATICS
, 2002
"... Suppose that f: [N] k → N. A set A ⊆ N is free for f if for all x1,..., xk ∈ A with x1 < x2 < · · · < xk, f(x1,..., xk) ∈ A implies f(x1,..., xk) ∈ {x1,..., xk}. The free set theorem asserts that every function f has an infinite free set. This paper addresses the computability theoreti ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
(Show Context)
Suppose that f: [N] k → N. A set A ⊆ N is free for f if for all x1,..., xk ∈ A with x1 < x2 < · · · < xk, f(x1,..., xk) ∈ A implies f(x1,..., xk) ∈ {x1,..., xk}. The free set theorem asserts that every function f has an infinite free set. This paper addresses the computability theoretic content and logical strength of the free set theorem. In particular, we prove that Ramsey’s theorem for pairs implies the free set theorem for pairs, and show that every computable f: [N] k → N has an infinite Π 0 k free set.