Separation and Weak König's Lemma

A. James Humphreys
Stephen G. Simpson

simpson@math.psu.edu
Pennsylvania State University
October 25, 1998

Simpson's research was partially supported by NSF grant DMS-9303478.

This paper has been accepted October 10, 1997 for publication in the Journal of Symbolic Logic.

Abstract:

We continue the work of [14,3,1,19,16,4,12,11,20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$. We show that the separation theorem for separably closed convex sets is equivalent to $\mathsf{ACA}_0$ over $\mathsf{RCA}_0$. Our strategy for proving these geometrical Hahn-Banach theorems is to reduce to the finite-dimensional case by means of a compactness argument.