Investigation of the Equivalence of the Axiom of Choice and Zorn’s Lemma from the Viewpoint of the Hierarchy of Types

Abstract

It is a well known fact that Zermelo’s Axiom of Choice and Zorn’s Lemma are equivalent logical assumptions. However, an investigation from the viewpoint of the hierarchy of types reveals a complication in the nature of this equivalence. In a type-theoretical formalism both postulates enter as spectra (ZA^α) and (ZL^α) of formulas, where ZA^α is Zermelo’s Axiom and ZL^α is Zorn’s Lemma stated for variables of a fixed type α. The complication has its origin in the fact that, although the formula ZA^α implies ZL^α, in order to deduce ZA^α it seems to be necessary to assert Zorn’s Lemma ZL^α for a type β which is higher than α. In other words, we don’t know a proof assuring the equivalence of the two formulas ZA^α and ZL^α. Therefore the equivalence of Zermelo’s Axiom and Zorn’s Lemma has to be understood in the sense that the assertion of the formulas ZA^α for all types α implies every one of the formulas ZL ^β and conversely the assertion of the spectrum of formulas (ZL^α) implies every one of the formulas ZA^β.