ON C⁽ⁿ⁾-EXTENDIBLE CARDINALS

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.