The effectivity of existential statements in axiomatic set theory

Abstract

Mathematical experience indicates a close connection between the axiom of choice and noneffective existence proofs. In order to make this connection a subject of a systematic study it seems appropriate to approach this problem from the point of view of the hierarchy developed by the author in [8]. It turns out, as can be seen from various related results and examples, that the connection between the axiom of choice and effectivity comes up when one considers sentences of the form (∀ x)(∃ y) χ( x, y), where, throughout this abstract, χ( x, y) is a formula which does not contain quantifiers other than quantifiers restricted to sets (such as (∀ t)( t ϵ s → …) or (∃ t)( t ϵ s …). Many statements of set theory are of this form; e.g. the ordering theorem, the Boolean prime ideal theorem, the generalized continuum hypothesis. If one can prove in the Zermelo-Fraenkel set theory without the axiom of choice ( ZF) that (∀ x)(∃ y) χ( x, y) then there is a term τ( u) of set theory such that ZF ⥼(∀x) (there is a finite subset u of the transitive closure of x) χ( x, τ( u)). One cannot expect to get a “more effective” solution for y since no such solution exists even for y in (∀ x)(∃ y)( x = 0 v y ϵ x). If one can prove (∀ x)(∃ y) χ( x, y) with the aid of the axiom of choice then there is a term τ such that ZF ⥼(∀x) (for every well-ordering r of the transitive closure of x) χ( x, τ( r)).