Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On arithmetical classifications of inaccessable cardinals and their applications
HTML articles powered by AMS MathViewer

by Géza Fodor and Attila Máté PDF
Trans. Amer. Math. Soc. 175 (1973), 69-99 Request permission

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.
References
  • Paul J. Cohen, Set theory and the continuum hypothesis, W. A. Benjamin, Inc., New York-Amsterdam, 1966. MR 0232676
  • G. Fodor, Eine Bemerkung zur Theorie der regressiven Funktionen, Acta Sci. Math. (Szeged) 17 (1956), 139–142 (German). MR 82450
  • G. Fodor, On a process concerning inaccessible cardinals. I, Acta Sci. Math. (Szeged) 27 (1966), 111–124. MR 200168
  • G. Fodor, On a process concerning inaccessible cardinals. II, Acta Sci. Math. (Szeged) 27 (1966), 129–140. MR 211868
  • G. Fodor, On a process concerning inaccessible cardinals. III, Acta Sci. Math. (Szeged) 28 (1967), 197–200. MR 216959
  • Haim Gaifman, A generalization of Malho’s method for obtaining large cardinal numbers, Israel J. Math. 5 (1967), 188–200. MR 221947, DOI 10.1007/BF02771107
  • W. Hanf, Incompactness in languages with infinitely long expressions, Fund. Math. 53 (1963/64), 309–324. MR 160732, DOI 10.4064/fm-53-3-309-324
  • 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 (1963/64), 225–308. MR 166107, DOI 10.4064/fm-53-3-225-308
  • Azriel Lévy, Axiom schemata of strong infinity in axiomatic set theory, Pacific J. Math. 10 (1960), 223–238. MR 124205
  • P. Mahlo, Über lineare transfinite Mengen, Ber. Verh. Königl. Sächs. Ges. Wiss. Leipzig Math.-Phys. K1. 63 (1911), 187-225. —, Zur Theorie und Anwendungen der $\delta$-Zahlen, Ber. Verh. Königl. Sächs. Ges. Wiss. Leipzig Math.-Phys. K1. 64 (1912), 108-112. —, Zur Theorie und Anwendungen der $\delta$-Zahlen, Ber. Verh. Königl. Sächs. Ges. Wiss. Leipzig Math.-Phys. K1. 65 (1913), 268-282.
  • Andrzej Mostowski, An undecidable arithmetical statement, Fund. Math. 36 (1949), 143–164. MR 35721, DOI 10.4064/fm-36-1-143-164
  • J. von Neumann, Zur Einführung der transfiniten Zahlen, Acta Sci. Math. (Szeged) 1 (1922/23), 199-208.
  • Walter Neumer, Verallgemeinerung eines Satzes von Alexandroff und Urysohn, Math. Z. 54 (1951), 254–261 (German). MR 43860, DOI 10.1007/BF01574826
  • J. C. Shepherdson, Inner models for set theory. I, J. Symbolic Logic 16 (1951), 161–190. MR 45073, DOI 10.2307/2266389
  • A. Tarski, Remarks on predicate logic with infinitely long expressions, Colloq. Math. 6 (1958), 171–176. MR 99915, DOI 10.4064/cm-6-1-171-176
  • 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
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 02K35, 04A10
  • Retrieve articles in all journals with MSC: 02K35, 04A10
Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 175 (1973), 69-99
  • MSC: Primary 02K35; Secondary 04A10
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0323569-2
  • MathSciNet review: 0323569