Abstract
AbstractLet X be a set, and let be the superstructure of X, where X0 = X and is the power set of X) for n ∈ ω. The set X is called a flat set if and only if for each x ∈ X, and x ∩ ŷ = ø for x, y ∈ X such that x ≠ y. where is the superstructure of y. In this article, it is shown that there exists a bijection of any nonempty set onto a flat set. Also, if is an ultrapower of (generated by any infinite set I and any nonprincipal ultrafilter on I), it is shown that is a nonstandard model of X: i.e., the Transfer Principle holds for and , if X is a flat set. Indeed, it is obvious that is not a nonstandard model of X when X is an infinite ordinal number. The construction of flat sets only requires the ZF axioms of set theory. Therefore, the assumption that X is a set of individuals (i.e., x ≠ ϕ and a ∈ x does not hold for x ∈ X and for any element a) is not needed for to be a nonstandard model of X.
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