Rado's conjecture and its Baire version

Rado's Conjecture is a compactness/reflection principle that says any nonspecial tree of height $\omega_1$ has a nonspecial subtree of size $\leq \aleph_1$. Though incompatible with Martin's Axiom, Rado's Conjecture turns out to have many interesting consequences that are consequences of forcing axioms. In this paper, we obtain consistency results concerning Rado's Conjecture and its Baire version. In particular, we show a fragment of PFA, that is the forcing axiom for \emph{Baire Indestructibly proper forcings}, is compatible with the Baire Rado's Conjecture. As a corollary, Baire Rado's Conjecture does not imply Rado's Conjecture. Then we discuss the strength and limitations of the Baire Rado's Conjecture regarding its interaction with simultaneous stationary reflection and some families of weak square principles. Finally we investigate the influence of the Rado's Conjecture on some polarized partition relations.Comment: Incorporated comments and corrections from the refere

Two-cardinal ideal operators and indescribability

A well-known version of Rowbottom's theorem for supercompactness ultrafilters leads naturally to notions of two-cardinal Ramseyness and corresponding normal ideals introduced herein. Generalizing results of Baumgartner [7, 8], Feng [22] and the first author [16, 17], we study the hierarchies associated with a particular version of two-cardinal Ramseyness and a strong version of two-cardinal ineffability, as well as the relationships between these hierarchies and a natural notion of transfinite two-cardinal indescribability.Comment: Fixed more typos in the first paragrap

Characterizations of the weakly compact ideal on PκλP_\kappa\lambda

Hellsten \cite{MR2026390} gave a characterization of $\Pi^1_n$-indescribable subsets of a $\Pi^1_n$-indescribable cardinal in terms of a natural filter base: when $\kappa$ is a $\Pi^1_n$-indescribable cardinal, a set $S\subseteq\kappa$ is $\Pi^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq \kappa$. We generalize Hellsten's characterization to $\Pi^1_n$-indescribable subsets of $P_\kappa\lambda$, which were first defined by Baumgartner. After showing that under reasonable assumptions the $\Pi^1_0$-indescribability ideal on $P_\kappa\lambda$ equals the minimal \emph{strongly} normal ideal $\text{NSS}_{\kappa,\lambda}$ on $P_\kappa\lambda$, and is not equal to $\text{NS}_{\kappa,\lambda}$ as may be expected, we formulate a notion of $n$-club subset of $P_\kappa\lambda$ and prove that a set $S\subseteq P_\kappa\lambda$ is $\Pi^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq P_\kappa\lambda$. We also prove that elementary embeddings considered by Schanker \cite{MR2989393} witnessing \emph{near supercompactness} lead to the definition of a normal ideal on $P_\kappa\lambda$, and indeed, this ideal is equal to Baumgartner's ideal of non--$\Pi^1_1$-indescribable subsets of $P_\kappa\lambda$. Additionally, as applications of these results we answer a question of Cox-L\"ucke \cite{MR3620068} about $\mathcal{F}$-layered posets, provide a characterization of $\Pi^m_n$-indescribable subsets of $P_\kappa\lambda$ in terms of generic elementary embeddings, prove several results involving a two-cardinal weakly compact diamond principle and observe that a result of Pereira \cite{MR3640048} yeilds the consistency of the existence of a $(\kappa,\kappa^+)$-semimorasses $\mu\subseteq P_\kappa\kappa^+$ which is $\Pi^1_n$-indescribable for all $n<\omega$.Comment: revised version for APA

Subtle and Ineffable Tree Properties

In the style of the tree property, we give combinatorial principles that capture the concepts of the so-called subtle and ineffable cardinals in such a way that they are also applicable to small cardinals. Building upon these principles we then develop a further one that even achieves this for supercompactness. We show the consistency of these principles starting from the corresponding large cardinals. Furthermore we show the equiconsistency for subtle and ineffable. For supercompactness, utilizing the failure of square we prove that the best currently known lower bounds for consistency strength in general can be applied. The main result of the thesis is the theorem that the Proper Forcing Axiom implies the principle corresponding to supercompactness.In Anlehnung an die Baumeigenschaft geben wir kombinatorische Prinzipien an, die die Konzepte der sogenannten subtle und ineffable Kardinalzahlen so einfangen, dass diese auch für kleine Kardinalzahlen anwendbar sind. Auf diesen Prinzipien aufbauend entwickeln wir dann ein weiteres, das dies sogar für superkompakte Kardinalzahlen leistet. Wir zeigen die Konsistenz dieser Prinzipien ausgehend von den jeweils entsprechenden großen Kardinalzahlen. Zudem zeigen wir die Äquikonsistenz für subtle und ineffable. Für Superkompaktheit beweisen wir durch das Fehlschlagen des Quadratprinzips, dass die besten derzeit bekannten unteren Schranken für Konsistenzstärke anwendbar sind. Das Hauptresultat der Arbeit ist das Ergebnis, dass das Proper Forcing Axiom das der Superkompaktheit entsprechende Prinzip impliziert

Embeddings into outer models

We explore the possibilities for elementary embeddings $j : M \to N$, where $M$ and $N$ are models of ZFC with the same ordinals, $M \subseteq N$, and $N$ has access to large pieces of $j$. We construct commuting systems of such maps between countable transitive models that are isomorphic to various canonical linear and partial orders, including the real line $\mathbb R$

Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

On constructions with 22-cardinals

We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman's neat simplified morasses called $2$-cardinals. The theory of related trees, gaps, colorings of pairs and forcing notions is reformulated and sketched from a unifying point of view with the focus on the applicability to constructions of mathematical structures like Boolean algebras, Banach spaces or compact spaces. A new result which we obtain as a side product is the consistency of the existence of a function $f:[\lambda^{++}]^2\rightarrow[\lambda^{++}]^{\leq\lambda}$ with the appropriate $\lambda^+$-version of property $\Delta$ for regular $\lambda\geq\omega$ satisfying $\lambda^{<\lambda}=\lambda$.Comment: Minor correction

Towers and clubs

We revisit several results concerning club principles and nonsaturation of the nonstationary ideal, attempting to improve them in various ways. So we typically deal with a (non necessarily normal) ideal $J$ extending the nonstationary ideal on a regular uncountable (non necessarily successor) cardinal $\kappa$, our goal being to witness the nonsaturation of $J$ by the existence of towers (of length possibly greater than $\kappa^+$)

Small embeddings, forcing with side conditions, and large cardinal characterizations

In this thesis, we provide new characterizations for several well-studied large cardinal notions. These characterizations will be of two types. Motivated by seminal work of Magidor, the first type characterizes large cardinals through the existence of so-called small embeddings, elementary embeddings between set-sized structures that map their critical point to the large cardinal in question. Building up on these characterizations, we also provide characterizations of many large cardinal notions through the validity of certain combinatorial principles at omega_2 in generic extensions. The combinatorial principles used in these characterizations are generalizations of large cardinal properties defined through small embeddings that can also hold at accessible cardinals and, for inaccessible cardinals, these principles are equivalent to the original large cardinal property. In this thesis, we focus on generic extensions obtained via the pure side condition forcing introduces by Neeman in his studies of forcing axioms and their generalizations. Our results will provide these two types of characterizations for some of the most prominent large cardinal notions, including inaccessible, Mahlo, Pi^m_n-indescribable cardinals, subtle, lambda-ineffable, and supercompact cardinals. In addition, we will derive small embedding characterizations of measurable, lambda-supercompact and huge cardinals, as well as forcing characterizations of almost huge and super almost huge cardinals. As an application of techniques developed in this work, we provide new proofs of Weiß 's results on the consistency strength of generalized tree properties, eliminating problematic arguments contained in his original proofs. The work presented in this thesis is joint work with Peter Holy and Philipp Lücke. It will be published in the following papers: Peter Holy, Philipp Lücke and Ana Njegomir. Small Embedding Characterizations for Large Cardinals. Annals of Pure and Applied Logic. Volume 170, Issue 2, pp. 251-271, 2019. Peter Holy, Philipp Lücke and Ana Njegomir. Characterizing large cardinals through Neeman's pure side condition forcing. Submitted to Fundamenta Mathematicae, 28 pages, 2018