Long reals

The familiar continuum R of real numbers is obtained by a well-known procedure which, starting with the set of natural numbers N=\omega, produces in a canonical fashion the field of rationals Q and, then, the field R as the completion of Q under Cauchy sequences (or, equivalently, using Dedekind cuts). In this article, we replace \omega by any infinite suitably closed ordinal \kappa in the above construction and, using the natural (Hessenberg) ordinal operations, we obtain the corresponding field \kappa-R, which we call the field of the \kappa-reals. Subsequently, we study the properties of the various fields \kappa-R and develop their general theory, mainly from the set-theoretic perspective. For example, we investigate their connection with standard themes such as forcing and descriptive set theory

Superstrong and other large cardinals are never Laver indestructible

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, \Sigma_n-reflecting cardinals, \Sigma_n-correct cardinals and \Sigma_n-extendible cardinals (all for n>2) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if \kappa\ exhibits any of them, with corresponding target \theta, then in any forcing extension arising from nontrivial strategically <\kappa-closed forcing Q in V_\theta, the cardinal \kappa\ will exhibit none of the large cardinal properties with target \theta\ or larger.Comment: 19 pages. Commentary concerning this article can be made at http://jdh.hamkins.org/superstrong-never-indestructible. Minor changes in v

Large cardinals and resurrection axioms

In the current dissertation we work in set theory and we study both various large cardinal hierarchies and issues related to forcing axioms and generic absoluteness. The necessary preliminaries may be found, as it should be anticipated, in the first chapter. In Chapter 2, we study several C(n) - cardinals as introduced by J. Bagaria (cf. [1]). In the context of an elementary embedding associated with some fixed C(n) - cardinal, and under adequate assumptions, we derive consistency (upper) bounds for the large cardinal notion at hand; in particular, we deal with the C(n) - versions of tallness, superstrongness, strongness, supercompactness, and extendibility. As far as the two latter notions are concerned, we further study their connection, giving an equivalent formulation of extendibility as well. We also consider the cases of C(n) -Woodin and of C(n) – strongly compact cardinals which were not studied in [1] and we get characterizations for them in terms of their ordinary counterparts. In Chapter 3, we briefly discuss the interaction of C(n) – cardinals with the forcing machinery, presenting some applications of ordinary techniques. In Chapter 4, we turn our attention to extendible cardinals; by a combination of methods and results from Chapter 2, we establish the existence of apt Laver functions for them. Although the latter was already known (cf. [2]), it is proved from a fresh viewpoint, one which nicely ties with the material of Chapter 5. We also argue that in the case of extendible cardinals one cannot use such Laver functions in order to attain indestructibility results. Along the way, we give an additional characterization of extendibility, and we, moreover, show that the global GCH can be forced while preserving such cardinals. In Chapter 5, we focus on the resurrection axioms as they are introduced by J.D. Hamkins and T. Johnstone (cf. [3]). Initially, we consider the class of stationary preserving posets and, assuming the (consistency of the) existence of an extendible cardinal, we obtain a model in which the resurrection axiom for this class holds. By analysing the proof of the previous result, we are led to much stronger forms of resurrection for which we introduce a family of axioms under the general name “Unbounded Resurrection”. We then prove that the consistency of these axioms follows from that of (the existence of) an extendible cardinal and that, for the appropriate classes of posets, they are strengthenings of the forcing axioms PFA and MM. We furthermore consider several implications of the unbounded resurrection axioms (e.g., their effect on the continuum, for the classes of c.c.c. and of sygma- closed posets) together with their connection with the corresponding ones of [3]. Finally, we also establish some consistency lower bounds for such axioms, mainly by deriving failures of (weak versions of) squares. We conclude our current mathematical quest with a few final remarks and a small list of open questions, followed by an Appendix on extenders and (some of) their applications. References [1] Bagaria, J., C (n)–cardinals. In Archive Math. Logic, Vol. 51 (3–4), pp. 213–240, 2012. [2] Corazza, P., Laver sequences for extendible and super–almost–huge cardinals. In J. Symbolic Logic, Vol. 64 (3), pp. 963–983, 1999. [3] Johnstone, T., Notes to “The Resurrection Axioms”. Unpublished notes (2009)