Academic literature on the topic 'Large cardinals (Mathematics)'

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory. §0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture: The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets Vα indexed by the ordinal numbers. The recursive definition reads: (where } is the power set of x); Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture.

S. Argyros and N. Kalamidas([l], repeated in [2], Theorem 6·15) proved the following. If κ is a cardinal of uncountable cofinality, and 〈Eξ〉ξ<κ is a family of measurable sets in a probability space (X, μ) such that infξ<κμEξ = δ, and if n ≥ 1, , then there is a set Γ ⊆ κ such that #(Γ) = κ and μ(∩ξ∈IEξ) ≥ γ whenever I ⊆ ξ has n members. In Proposition 7 below I refine this result by (i) taking any γ < δn (which is best possible) and (ii) extending the result to infinite cardinals of countable cofinality, thereby removing what turns out to be an irrelevant restriction. The proof makes it natural to perform a further extension to general uniformly bounded families of non-negative measurable functions in place of characteristic functions.

AbstractWe investigate the possibilities of global versions of Chang’s Conjecture that involve singular cardinals. We show some $$\mathrm{ZFC} $$ ZFC limitations on such principles and prove relative to large cardinals that Chang’s Conjecture can consistently hold between all pairs of limit cardinals below $$\aleph _{\omega ^\omega }$$ ℵ ω ω .

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).

The present dissertation is a contribution to the field of Mathematical Logic and, more particularly, to the subfield of Set Theory. Within Set theory, we are mainly concerned with the interactions between the largecardinal axioms and the method of Forcing. This is the line of research with a deeper impact in the subsequent configuration of modern Mathematics. This area has found many central applications in Topology [ST71][Tod89], Algebra [She74][MS94][DG85][Dug85], Analysis [Sol70] or Category Theory [AR94][Bag+15], among others. The dissertation is divided in two thematic blocks: In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopenka’s Principle (Part I). In Block II we make a contribution to Singular Cardinal Combinatorics (Part II and Part III). Specifically, in Part I we investigate the Identity Crisis phenomenon in the region comprised between the first supercompact cardinal and Vopenka’s Principle. As a result, we settle all the questions that were left open in [Bag12, §5]. Afterwards, we present a general theory of preservation of C(n)– extendible cardinals under class forcing iterations from which we derive many applications. In Part II and Part III we analyse the relationship between the Singular Cardinal Hypothesis (SCH) and other combinatorial principles, such as the tree property or the reflection of stationary sets. In Part II we generalize the main theorems of [FHS18] and [Sin16] and manage to weaken the largecardinal hypotheses necessary for Magidor-Shelah’s theorem [MS96]. Finally, in Part III we introduce the concept of _-Prikry forcing as a generalization of the classical notion of Prikry-type forcing. Subsequently we devise an abstract iteration scheme for this family of posets and, as an application, we prove the consistency of ZFC + ¬SCH_ + Refl(La present tesi és una contribució a l’estudi de la Lògica Matemàtica i més particularment a la Teoria de Conjunts. Dins de la Teoria de Conjunts, la nostra àrea de recerca s’emmarca dins l’estudi de les interaccions entre els Axiomes de Grans Cardinals i el mètode de Forcing. Aquestes dues eines han tigut un impacte molt profund en la configuració de la matemàtica contemporànea com a conseqüència de la resolució de qüestions centrals en Topologia [ST71][Tod89], Àlgebra [She74][MS94][DG85][Dug85], Anàlisi Matemàtica [Sol70] o Teoria de Categories [AR94][Bag+15], entre d’altres. La tesi s’articula entorn a dos blocs temàtics. Al Bloc I analitzem la jerarquia de Grans Cardinals compresa entre el primer cardinal supercompacte i el Principi de Vopenka (Part I), mentre que al Bloc II estudiem alguns problemes de la Combinatòria Cardinal Singular (Part II i Part III). Més precisament, a la Part I investiguem el fenòmen de Crisi d’Identitat en la regió compresa entre el primer cardinal supercompacte i el Principi de Vopenka. Com a conseqüència d’aquesta anàlisi resolem totes les preguntes obertes de [Bag12, §5]. Posteriorment presentem una teoria general de preservació de cardinals C(n)–extensibles sota iteracions de longitud ORD, de la qual en derivem nombroses aplicacions. A la Part II i Part III analitzem la relació entre la Hipòtesi dels Cardinals Singulars (SCH) i altres principis combinatoris, tals com la Propietat de l’Arbre o la reflexió de conjunts estacionaris. A la Part II obtenim sengles generalitzacions dels teoremes principals de [FHS18] i [Sin16] i afeblim les hipòtesis necessàries perquè el teorema de Magidor-Shelah [MS96] siga cert. Finalment, a la Part III, introduïm el concepte de forcing _-Prikry com a generalització de la noció clàssica de forcing del tipus Prikry. Posteriorment dissenyem un esquema d’iteracions abstracte per aquesta família de forcings i, com a aplicació, derivem la consistència de ZFC + ¬SCH_ + Refl(