Extension methods in cardinal arithmetic

Abstract

Functions (relations) defined on the nonnegative integers are extended to the cardinal numbers by the method of Myhill (Nerode) respectively. We obtain various results relating these extensions and conclude with an analysis of AE Horn sentences interpreted in the cardinal numbers. Let A \mathfrak {A} be the sentence ( ∀ x 1 ) ⋯ ( ∀ x n ) ( ∃ ! y ) b (\forall {x_1}) \cdots (\forall {x_n})(\exists !y)\mathfrak {b} where quantifiers are restricted to the Dedekind cardinals and b \mathfrak {b} is an equation built up from functors for cardinal addition, multiplication, and integer constants. One of our principal results is that A \mathfrak {A} is a theorem of set theory (with the axiom of choice replaced by the axiom of choice for sets of finite sets) if and only if we can prove that the uniquely determined Skolem function for A \mathfrak {A} extends an almost combinatorial function.