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Transactions of the American Mathematical Society

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On arithmetical classifications of inaccessable cardinals and their applications

Authors: Géza Fodor and Attila Máté
Journal: Trans. Amer. Math. Soc. 175 (1973), 69-99
MSC: Primary 02K35; Secondary 04A10
MathSciNet review: 0323569
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Abstract: Lately several authors, among them Fodor, Gaifman, Hanf, Keisler, Lévy and Tarski, dug out an interesting and unduly forgotten operation of Mahlo that, loosely speaking, from a sequence of ordinals discards all those that are easy to locate in this sequence. The purpose of these authors was to invent strengthenings and schemes for repetitions of this and similar operations and to study the properties of cardinals that can be discarded in this way when started with a specific class; for example, the class of all inaccessible cardinals. Our attempt here is to consider such schemes for repetitions of operations that can in a sense be described in an arithmetical way, which might also be called constructive; our investigations are akin to the problem of constructive description of possibly large segments of, say, the set of all countable ordinals. Some applications of our classifications scheme are exhibited, questions ranging from definability of inaccessible cardinals in terms of sets of lower ranks to incompactness theorems in infinitary languages. The paper is concluded with an algebraic-axiomatic type study of our scheme.

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  • [1] Paul J. Cohen, Set theory and the continuum hypothesis, W. A. Benjamin, Inc., New York-Amsterdam, 1966. MR 0232676
  • [2] G. Fodor, Eine Bemerkung zur Theorie der regressiven Funktionen, Acta Sci. Math. Szeged 17 (1956), 139–142 (German). MR 0082450
  • [3] G. Fodor, On a process concerning inaccessible cardinals. I, Acta Sci. Math. (Szeged) 27 (1966), 111–124. MR 0200168
  • [4] G. Fodor, On a process concerning inaccessible cardinals. II, Acta Sci. Math. (Szeged) 27 (1966), 129–140. MR 0211868
  • [5] G. Fodor, On a process concerning inaccessible cardinals. III, Acta Sci. Math. (Szeged) 28 (1967), 197–200. MR 0216959
  • [6] Haim Gaifman, A generalization of Malho’s method for obtaining large cardinal numbers, Israel J. Math. 5 (1967), 188–200. MR 0221947,
  • [7] W. Hanf, Incompactness in languages with infinitely long expressions, Fund. Math. 53 (1963/1964), 309–324. MR 0160732
  • [8] H. J. Keisler and A. Tarski, From accessible to inaccessible cardinals. Results holding for all accessible cardinal numbers and the problem of their extension to inaccessible ones, Fund. Math. 53 (1963/1964), 225–308. MR 0166107
  • [9] Azriel Lévy, Axiom schemata of strong infinity in axiomatic set theory, Pacific J. Math. 10 (1960), 223–238. MR 0124205
  • [10] P. Mahlo, Über lineare transfinite Mengen, Ber. Verh. Königl. Sächs. Ges. Wiss. Leipzig Math.-Phys. K1. 63 (1911), 187-225.
  • [11] -, Zur Theorie und Anwendungen der $ \delta $-Zahlen, Ber. Verh. Königl. Sächs. Ges. Wiss. Leipzig Math.-Phys. K1. 64 (1912), 108-112.
  • [12] -, Zur Theorie und Anwendungen der $ \delta $-Zahlen, Ber. Verh. Königl. Sächs. Ges. Wiss. Leipzig Math.-Phys. K1. 65 (1913), 268-282.
  • [13] Andrzej Mostowski, An undecidable arithmetical statement, Fund. Math. 36 (1949), 143–164. MR 0035721
  • [14] J. von Neumann, Zur Einführung der transfiniten Zahlen, Acta Sci. Math. (Szeged) 1 (1922/23), 199-208.
  • [15] Walter Neumer, Verallgemeinerung eines Satzes von Alexandroff und Urysohn, Math. Z. 54 (1951), 254–261 (German). MR 0043860,
  • [16] J. C. Shepherdson, Inner models for set theory. I, J. Symbolic Logic 16 (1951), 161–190. MR 0045073,
  • [17] A. Tarski, Remarks on predicate logic with infinitely long expressions, Colloq. Math. 6 (1958), 171–176. MR 0099915
  • [18] Alfred Tarski, Some problems and results relevant to the foundations of set theory, Logic, Methodology and Philosophy of Science (Proc. 1960 Internat. Congr.), Stanford Univ. Press, Stanford, Calif., 1962, pp. 125–135. MR 0151397

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Additional Information

Keywords: $ \alpha $-complete ideal, $ \alpha $-truncation of the process, arithmetical classifications of cardinals, band, canonical process, canonical sequence, definability of inaccessible cardinals, divergent function, dropping-out, fixed-point operation, Fodor's process, hyper-inaccessible cardinal, Mahlo's process, measurable cardinal, nonmingling property, pseudo-universe, recollection step, regressive function, stationary-point operation, stationary set, strongly incompact cardinal, weakly compact cardinal
Article copyright: © Copyright 1973 American Mathematical Society