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ON C(n)-EXTENDIBLE CARDINALS

Published online by Cambridge University Press:  23 October 2018

KONSTANTINOS TSAPROUNIS
KONSTANTINOS TSAPROUNIS*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF THE AEGEAN SAMOS, GREECEE-mail:kostas.tsap@gmail.com
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Abstract

The hierarchies of C(n)-cardinals were introduced by Bagaria in [1] and were further studied and extended by the author in [18] and in [20]. The case of C(n)-extendible cardinals, and of their C(n)+-extendibility variant, is of particular interest since such cardinals have found applications in the areas of category theory, of homotopy theory, and of model theory (see [2], [3], and [4], respectively). However, the exact relation between these two notions had been left unclarified. Moreover, the question of whether the Generalized Continuum Hypothesis (GCH) can be forced while preserving C(n)-extendible cardinals (for n1) also remained open. In this note, we first establish results in the direction of exactly controlling the targets of C(n)-extendibility embeddings. As a corollary, we show that every C(n)-extendible cardinal is in fact C(n)+-extendible; this, in turn, clarifies the assumption needed in some applications obtained in [3]. At the same time, we underline the applicability of our arguments in the context of C(n)-ultrahuge cardinals as well, as these were introduced in [20]. Subsequently, we show that C(n)-extendible cardinals carry their own Laver functions, making them the first known example of C(n)-cardinals that have this desirable feature. Finally, we obtain an alternative characterization of C(n)-extendibility, which we use to answer the question regarding forcing the GCH affirmatively.

Keywords

03E5503E35extendible cardinalsC(n)-cardinalsLaver functionsultrahuge cardinalsGeneralized Continuum Hypothesis (GCH)
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 3 , September 2018 , pp. 1112 - 1131
Copyright
Copyright © The Association for Symbolic Logic 2018 

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