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〉 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 υ 0ϵω. (2) If ξ T = 〈 a t' > t〉 tϵT is a nonstandard exchange economy, then T ≊ {0,1… ,K} for some standard finite K<ω. Thus no nontrivial nonstandard exchange economy: ξ T = lang; a ̄ t' > t〉 tϵT exists. (3) There exists an infinite class of market games: { Γ j =〈 X, A( X), υ j 〉} j< ω all of which have empty cores. This theorem shows that the verification of Edgeworth's Conjecture by the Aumann-Hildenbrand and Brown-Robinson models cannot be proved in M[G̃] and thus are independent of ZF; e.g., from (3) an arbitrary market game need not have a nonempty core and from (2) infinitessimal traders simply do not exist. By way of contrast, we prove that Arrow's Impossibility Theorem for infinite societies M = 〈ω,X,P, F 〉 is provable within M[G̃] as are major theorems of Quantum Mechanics in physics.