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A choice free theory of Dedekind cardinals1

  • Erik Ellentuck (a1)


In this paper we continue our investigation of the Dedekind cardinals which was initiated in [2]. Those results are summarized below. Let ω be the finite cardinals and Δ the Dedekind cardinals. In [7] Myhill defined a class of functions ƒ: Χκω→ω, which he called the combinatorial functions, and which he applied to the study of recursive equivalence types.



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Research for this paper was supported in part by National Science Foundation contract number GP 5786.



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[1]Cohen, P. J., Set theory and the continuum hypothesis, Benjamin, New York, 1966.
[2]Ellentuck, E., The universal properties of Dedekind finite cardinals, Annals of Mathematics, vol. 82 (1965), pp. 225248.
[3]Gödel, K., Consistency proof for the generalized continuum hypothesis, Proceedings of the National Academy of Sciences of the U.S.A., vol. 25 (1939), pp. 220224.
[4]Gödel, K., The consistency of the continuum hypothesis, Princeton Univ. Press, Princeton, N.J., 1958.
[5]Lévy, A., Independence results in set theory by Cohen's method. I, II, III, IV, Notices of the American Mathematical Society, vol. 10 (1963), pp. 592593.
[6]Mostowski, A., Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip, Fundamenta Mathematicae, vol. 32 (1939), pp. 201252.
[7]Myhill, J., Recursive equivalence types and combinatorial functions, Bulletin of the American Mathematical Society, vol. 64 (1958), pp. 373376.
[8]Shoenfield, J., The problem of predicativity, Essays on the foundations of mathematics, Magnes Press, Jerusalem, 1961, pp. 132139.


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