On the independence of core-equivalence results from Zermelo-Fraenkel set theory

Abstract

In this paper we prove that there exist consistent mathematical frameworks for the ordinary mathematics of Quantum Mechanics within ZF set theory for which the core equivalence results within abstract neoclassical mathematical economics are not provable. More precisely, we use Cohen-forcing to prove the following results with respect to Zermelo-Fraenkel set theory (ZF): Theorem. There exists a model of ZF, M = 〈V,ε〉 and a set of forcing conditions P⊂B, for B a complete Boolean Algebra in M such that if G̃ is an M-generic ultrafilter on B, then M[ G ̃] = 〈V[ G ̃],ε{lunate}〉 is a model of ZF for which the following statements are true: (1) Every countable society: M = 〈ω,X,P,F 〉 possesses an Arrowian Dictator υ0ε{lunate}ω. (2) If ξT = 〈at' > t〉tε{lunate}T is a nonstandard exchange economy, then T ≊ {0,1...,K} for some standard finite K t〉tε{lunate}T exists. (3) There exists an infinite class of market games: {Γj=〈X,A(X),υj〉}j