TY - JOUR
AU - Lembcke, Jörn
AB - TWO EXTENSION THEOREMS EFFECTIVELY EQUIVALENT TO THE AXIOM OF CHOICE JORN LEMBCKE In this paper we show that Andenaes' version of the Hahn-Banach Theorem as well as a theorem on the existence of preimages of regular Borel measures effectively imply the axiom of multiple choice which, by a result of Feigner and Jech, is equivalent to the axiom of choice in Zermelo-Fraenkel set theory. Throughout this paper we will assume the Zermelo-Fraenkel axioms of set theory (ZF) excluding the axiom of choice (AC). 1. It is well-known (cf. Pincus [10]) that the Hahn-Banach Theorem cannot be proved in (ZF), however, it is strictly weaker than the full axiom of choice. On the other hand, there are theorems in functional analysis which are effectively equivalent to (AC) (cf. Bell and Fremlin [2] and Edwards [5]). I n the first part of this note we will consider a stronger version of the Hahn-Banach Theorem which will also turn out to be equivalent to the axiom of choice: ANDENAES' THEOREM (cf. [l])t- Let F be a linear subspace of a real vector space E and let S be a subset of E. Moreover, suppose that p : E -*• U is a 
TI - Two Extension Theorems Effectively Equivalent to the Axiom of Choice
JF - Bulletin of the London Mathematical Society
DO - 10.1112/blms/11.3.285
DA - 1979-10-01
UR - https://www.deepdyve.com/lp/wiley/two-extension-theorems-effectively-equivalent-to-the-axiom-of-choice-JGMerigxZt
SP - 285
EP - 288
VL - 11
IS - 3
DP - DeepDyve
ER -