The cardinality of powersets in finite models of the powerset axiom.
Alexander Abian, Wael A. Amin · Notre Dame Journal of Formal Logic · 1991
It is shown that in a finite model of the set-theoretical Powerset axiom a set s and its powerset (P(s) have the same number of elements.Additional results are also derived.Let (^e)bea finite model of the set-theoretical Powerset axiom, i.e., in (F,e) every set has a powerset.For instance, let us consider the finite model (M, e) whose domain consists of the four sets a,b,c,d and where the e-relation is defined by:(1) a=lb], b={a}, c={a 9 b,c} 9 d={a,b,c,d}.It can be readily verified that (M, e) is a model of the Powerset axiom.To this end, we have only to verify that every one of the sets α, b, c, d of the model (M,e) has a powerset in (M,e).For instance, to show that the powerset (P(c) of c exists in (M, e), we must show that all the subsets of c which exist in (M, e) are collected by a set of (M,e).As (1) shows, c = [a,b,c] and therefore, from the point of view of the standard ZF set theory, c has 2 3 = 8 subsets given by: 0, {a} 9 [b} 9 {c}, {a,b}, {a,c}, {b,c}, {a,b,c}.On the other hand, as (1)