Extension methods in cardinal arithmetic
Author:
Erik Ellentuck
Journal:
Trans. Amer. Math. Soc. 149 (1970), 307325
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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 R. Bradford, Undecidability of the theory of Dedekind cardinal addition, Summer Institute on Axiomatic Set Theory, (Los Angeles, 1967) Amer. Math. Soc., Providence, R. I.
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 P. J. Cohen, Set theory and the continuum hypothesis, Benjamin, New York, 1966. MR 38 #999. MR 0232676 (38:999)
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 J. C. E. Dekker, A nonconstructive extension of the number system, J. Symbolic Logic 20 (1955). 204205.
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 , Almost combinatorial functions, Notices Amer. Math. Soc. 15 (1968), 650651. Abstract #68T500.
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 J. Myhill, Recursive equivalence types and combinatorial functions, Bull. Amer. Math. Soc. 64 (1958), 373376. MR 21 #7. MR 0101194 (21:7)
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 A. Nerode, Extensions to isols, Ann. of Math. (2) 73 (1961), 362403. MR 24 #A1215. MR 0131363 (24:A1215)
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 , Nonlinear combinatorial functions of isols, Math. Z. 86 (1965), 410424. MR 34 #5672. MR 0205846 (34:5672)
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 A. Tarski, Cancellation laws in the arithmetic of cardinals, Fund. Math. 36 (1949), 7792. MR 11, 335. MR 0032710 (11:335b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197002568680
PII:
S 00029947(1970)02568680
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
