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ON $C^{(n)}$-EXTENDIBLE CARDINALS
| Content Provider | Scilit |
|---|---|
| Author | Tsaprounis, Konstantinos |
| Copyright Year | 2018 |
| Description | 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. |
| Related Links | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/B3C16161B6526301FF4273723A3D5295/S0022481218000312a.pdf/div-class-title-on-span-class-italic-c-span-span-class-sup-span-class-italic-n-span-span-extendible-cardinals-div.pdf |
| Ending Page | 1131 |
| Page Count | 20 |
| Starting Page | 1112 |
| ISSN | 00224812 |
| e-ISSN | 19435886 |
| DOI | 10.1017/jsl.2018.31 |
| Journal | The Journal of Symbolic Logic |
| Issue Number | 3 |
| Volume Number | 83 |
| Language | English |
| Publisher | Cambridge University Press (CUP) |
| Publisher Date | 2018-09-01 |
| Access Restriction | Open |
| Subject Keyword | The Journal of Symbolic Logic Extendible Cardinals Ultrahuge Cardinals Generalized Continuum Hypothesis (gch) |
| Content Type | Text |
| Resource Type | Article |
| Subject | Philosophy Logic |
