Indestructibility properties of remarkable cardinals

Cheng, Yong; Gitman, Victoria

doi:10.1007/s00153-015-0453-8

Cited by 2 publications

(1 citation statement)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us remark that, in the light of Corollary 3.5, it now follows that the assumption of C (n)+ -extendibility that has been used in various results appearing in [3] and elsewhere has now been rendered redundant. 7 The situation is thus clarified in the sense that we only need to consider C (n) -extendible cardinals, without any additional requirements on their witnessing embeddings, a notion that so far appears to be a rather robust and well-behaved one.…”

Section: (N)mentioning

confidence: 99%

ON C⁽ⁿ⁾-EXTENDIBLE CARDINALS

Tsaprounis¹

2018

J. symb. log.

View full text Add to dashboard Cite

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.

show abstract

Section: (N)mentioning

confidence: 99%