Extension methods in cardinal arithmetic

Author:
Erik Ellentuck

Journal:
Trans. Amer. Math. Soc. **149** (1970), 307-325

MSC:
Primary 02.60

MathSciNet review:
0256868

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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 be the sentence where quantifiers are restricted to the Dedekind cardinals and is an equation built up from functors for cardinal addition, multiplication, and integer constants. One of our principal results is that 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 extends an almost combinatorial function.

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DOI:
https://doi.org/10.1090/S0002-9947-1970-0256868-0

Keywords:
Dedekind cardinal,
universal cardinal,
combinatorial function,
eventually combinatorial function,
almost combinatorial function,
combinatorial operator,
frame,
Horn sentence

Article copyright:
© Copyright 1970
American Mathematical Society