Extension methods in cardinal arithmetic

Author:
Erik Ellentuck

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

MSC:
Primary 02.60

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

MathSciNet review:
0256868

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**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.**[2]**P. J. Cohen,*Set theory and the continuum hypothesis*, Benjamin, New York, 1966. MR**38**#999. MR**0232676 (38:999)****[3]**J. C. E. Dekker,*A non-constructive extension of the number system*, J. Symbolic Logic**20**(1955). 204-205.**[4]**E. Ellentuck,*The theory of Dedekind finite cardinals*, Dissertation, Univ. of California, Berkeley, Calif., 1962.**[5]**-,*The universal properties of Dedekind finite cardinals*, Ann. of Math. (2)**82**(1965), 225-248. MR**31**#4729. MR**0180494 (31:4729)****[6]**-,*The first order properties of Dedekind finite integers*, Fund. Math.**63**(1968), 7-25. MR**0238689 (39:53)****[7]**-,*A choice free theory of Dedekind cardinals*, J. Symbolic Logic**33**(1968), 60-84.**[8]**-,*Almost combinatorial functions*, Notices Amer. Math. Soc.**15**(1968), 650-651. Abstract #68T-500.**[9]**K. Gödel,*The consistency of the continuum hypothesis*, Ann. of Math. Studies, no. 3, Princeton Univ. Press, Princeton, N. J., 1940. MR**2**, 66. MR**0002514 (2:66c)****[10]**A. Lévy,*Independence results in set theory by Cohen's method*. I-IV, Notices Amer. Math. Soc.**10**(1963), 592-593. Abstract #63T-388.**[11]**J. Myhill,*Recursive equivalence types and combinatorial functions*, Bull. Amer. Math. Soc.**64**(1958), 373-376. MR**21**#7. MR**0101194 (21:7)****[12]**A. Nerode,*Extensions to isols*, Ann. of Math. (2)**73**(1961), 362-403. MR**24**#A1215. MR**0131363 (24:A1215)****[13]**-,*Extensions to isolic integers*, Ann. of Math. (2)**75**(1962), 419-448. MR**25**#3830. MR**0140410 (25:3830)****[14]**-,*Non-linear combinatorial functions of isols*, Math. Z.**86**(1965), 410-424. MR**34**#5672. MR**0205846 (34:5672)****[15]**A. Tarski,*Cancellation laws in the arithmetic of cardinals*, Fund. Math.**36**(1949), 77-92. MR**11**, 335. MR**0032710 (11:335b)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
02.60

Retrieve articles in all journals with MSC: 02.60

Additional Information

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