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)).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.