Contributions to the Theory of Large Cardinals through the Method of Forcing

The dissertation under comment is a contribution to the area of Set Theory concerned with the interactions between the method of Forcing and the so-called Large Cardinal axioms.The dissertation is divided into two thematic blocks. In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopěnka’s Principle (Part I). In turn, Block II is devoted to the investigation of some problems arising from Singular Cardinal Combinatorics (Part II and Part III).We commence Part I by investigating the Identity Crisis phenomenon in the region comprised between the first supercompact cardinal and Vopěnka’s Principle. As a result, we generalize Magidor’s classical theorems [2] to this higher region of the large-cardinal hierarchy. Also, our analysis allows to settle all the questions that were left open in [1]. Finally, we conclude Part I by presenting a general theory of preservation of $C^{(n)}$ -extendible cardinals under class forcing iterations. From this analysis we derive several applications. For instance, our arguments are used to show that an extendible cardinal is consistent with “ $(\lambda ^{+\omega })^{\mathrm {HOD}}<\lambda ^+$ , for every regular cardinal $\lambda $ .” In particular, if Woodin’s HOD Conjecture holds, and therefore it is provable in ZFC + “There exists an extendible cardinal” that above the first extendible cardinal every singular cardinal $\lambda $ is singular in HOD and $(\lambda ^+)^{\textrm {{HOD}}}=\lambda ^+$ , there may still be no agreement at all between V and HOD about successors of regular cardinals.In Part II and Part III we analyse the relationship between the Singular Cardinal Hypothesis (SCH) with other relevant combinatorial principles at the level of successors of singular cardinals. Two of these are the Tree Property and the Reflection of Stationary sets, which are central in Infinite Combinatorics.Specifically, Part II is devoted to prove the consistency of the Tree Property at both $\kappa ^+$ and $\kappa ^{++}$ , whenever $\kappa $ is a strong limit singular cardinal witnessing an arbitrary failure of the SCH. This generalizes the main result of [3] in two senses: it allows arbitrary cofinalities for $\kappa $ and arbitrary failures for the SCH.In the last part of the dissertation (Part III) we introduce the notion of $\Sigma $ -Prikry forcing. This new concept allows an abstract and uniform approach to the theory of Prikry-type forcings and encompasses several classical examples of Prikry-type forcing notions, such as the classical Prikry forcing, the Gitik-Sharon poset, or the Extender Based Prikry forcing, among many others.Our motivation in this part of the dissertation is to prove an iteration theorem at the level of the successor of a singular cardinal. Specifically, we aim for a theorem asserting that every $\kappa ^{++}$ -length iteration with support of size $\leq \kappa $ has the $\kappa ^{++}$ -cc, provided the iterates belong to a relevant class of $\kappa ^{++}$ -cc forcings. While there are a myriad of works on this vein for regular cardinals, this contrasts with the dearth of investigations in the parallel context of singular cardinals. Our main contribution is the proof that such a result is available whenever the class of forcings under consideration is the family of $\Sigma $ -Prikry forcings. Finally, and as an application, we prove that it is consistent—modulo large cardinals—the existence of a strong limit cardinal $\kappa $ with countable cofinality such that $\mathrm {SCH}_\kappa $ fails and every finite family of stationary subsets of $\kappa ^+$ reflects simultaneously.

Ordinal compactness

We introduce a new covering property, defined in terms of order types of sequences of open sets, rather than in terms of cardinalities. The most general form depends on two ordinal parameters. Ordinal compactness turns out to be a much more varied notion than cardinal compactness. We prove many nontrivial results of the form ?every [?,?]-compact topological space is [?',?']-compact?, for ordinals ?,?, ?'and ?' while only trivial results of the above form hold, if we restrict to regular cardinals. Counterexamples are provided showing that many results are optimal. Many spaces satisfy the very same cardinal compactness properties, but have a broad range of distinct behaviors, as far as ordinal compactness is concerned. A much more refined theory is obtained for T1 spaces, in comparison with arbitrary topological spaces. The notion of ordinal compactness becomes partly trivial for spaces of small cardinality.

LARGE CARDINALS BEYOND CHOICE

The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V (in the sense that it correctly computes successors of singular cardinals greater than δ) or HOD is “far” from V (in the sense that all regular cardinals greater than or equal to δ are measurable in HOD). The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that (assuming the existence of an extendible cardinal) that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails.

ON SINGULAR STATIONARITY II (TIGHT STATIONARITY AND EXTENDERS-BASED METHODS)

We study the notion of tightly stationary sets which was introduced by Foreman and Magidor in [8]. We obtain two consistency results showing that certain sequences of regular cardinals ${\langle {\kappa _n}\rangle _{n < \omega }}$ can have the property that in some generic extension, every ground-model sequence of fixed-cofinality stationary sets ${S_n} \subseteq {\kappa _n}$ is tightly stationary. The results are obtained using variations of the short-extenders forcing method.

SQUARES, ASCENT PATHS, AND CHAIN CONDITIONS

With the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević’s principle $\square \left( {\kappa ,\lambda } \right)$ implies an indexed version of $\square \left( {\kappa ,\lambda } \right)$, we show that for all infinite, regular cardinals $\lambda < \kappa$, the principle $\square \left( \kappa \right)$ implies the existence of a κ-Aronszajn tree containing a λ-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which ${\aleph _2}$-Aronszajn trees exist and all such trees contain ${\aleph _0}$-ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of $\square \left( \kappa \right)$.

TOWERS IN FILTERS, CARDINAL INVARIANTS, AND LUZIN TYPE FAMILIES

We investigate which filters onωcan contain towers, that is, a modulo finite descending sequence without any pseudointersection (in${[\omega ]^\omega }$). We prove the following results:(1)Many classical examples of nice tall filters contain no towers (in ZFC).(2)It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well).(3)It is consistent that all towers generate nonmeager filters (this answers a question of P. Borodulin-Nadzieja and D. Chodounský), in particular (consistently) Borel filters do not contain towers.(4)The statement “Every ultrafilter contains towers.” is independent of ZFC (this improves an older result of K. Kunen, J. van Mill, and C. F. Mills).Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters (${\rm{ad}}{{\rm{d}}^{\rm{*}}}\left( {\cal F} \right)$,${\rm{co}}{{\rm{f}}^{\rm{*}}}\left( {\cal F} \right)$,${\rm{no}}{{\rm{n}}^{\rm{*}}}\left( {\cal F} \right)$, and${\rm{co}}{{\rm{v}}^{\rm{*}}}\left( {\cal F} \right)$), and the existence of Luzin type families (of size$\ge {\omega _2}$), that is, if${\cal F}$is a filter then${\cal X} \subseteq {[\omega ]^\omega }$is an${\cal F}$-Luzin family if$\left\{ {X \in {\cal X}:|X \setminus F| = \omega } \right\}$is countable for every$F \in {\cal F}$.

On cuts in ultraproducts of linear orders I

For an ultrafilter [Formula: see text] on a cardinal [Formula: see text] we wonder for which pair [Formula: see text] of regular cardinals, we have: for any [Formula: see text]-saturated dense linear order [Formula: see text] has a cut of cofinality [Formula: see text] We deal mainly with the case [Formula: see text]