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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
DOI: https://doi.org/10.1090/S0002-9947-1973-0323569-2
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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DOI: https://doi.org/10.1090/S0002-9947-1973-0323569-2
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

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