Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Extension methods in cardinal arithmetic


Author: Erik Ellentuck
Journal: Trans. Amer. Math. Soc. 149 (1970), 307-325
MSC: Primary 02.60
MathSciNet review: 0256868
Full-text PDF Free Access

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 $ \mathfrak{A}$ be the sentence $ (\forall {x_1}) \cdots (\forall {x_n})(\exists !y)\mathfrak{b}$ where quantifiers are restricted to the Dedekind cardinals and $ \mathfrak{b}$ is an equation built up from functors for cardinal addition, multiplication, and integer constants. One of our principal results is that $ \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 $ \mathfrak{A}$ extends an almost combinatorial function.


References [Enhancements On Off] (What's this?)

  • [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] Paul J. Cohen, Set theory and the continuum hypothesis, W. A. Benjamin, Inc., New York-Amsterdam, 1966. MR 0232676
  • [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] Erik Ellentuck, The universal properties of Dedekind finite cardinals, Ann. of Math. (2) 82 (1965), 225–248. MR 0180494
  • [6] Erik Ellentuck, The first order properties of Dedekind finite integers, Fund. Math. 63 (1968), 7–25. MR 0238689
  • [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] Kurt Gödel, The Consistency of the Continuum Hypothesis, Annals of Mathematics Studies, no. 3, Princeton University Press, Princeton, N. J., 1940. MR 0002514
  • [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 0101194, 10.1090/S0002-9904-1958-10241-4
  • [12] Anil Nerode, Extensions to isols, Ann. of Math. (2) 73 (1961), 362–403. MR 0131363
  • [13] A. Nerode, Extensions to isolic integers, Ann. of Math. (2) 75 (1962), 419–448. MR 0140410
  • [14] A. Nerode, Non-linear combinatorial functions of isols, Math. Z. 86 (1965), 410–424. MR 0205846
  • [15] Alfred Tarski, Cancellation laws in the arithmetic of cardinals, Fund. Math. 36 (1949), 77–92. MR 0032710

Similar Articles

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