On Some Properties of Shelah Cardinals

We present several results concerning Shelah cardinals

An Easton like theorem in the presence of Shelah Cardinals

We show that Shelah cardinals are preserved under the canonical $GCH$ forcing notion. We also show that if $GCH$ holds and $F:REG\rightarrow CARD$ is an Easton function which satisfies some weak properties, then there exists a cofinality preserving generic extension of the universe which preserves Shelah cardinals and satisfies $\forall \kappa\in REG,~ 2^{\kappa}=F(\kappa)$. This gives a partial answer to a question asked by Cody [1] and independently by Honzik [5]. We also prove an indestructibility result for Shelah cardinals

The large cardinals between supercompact and almost-huge

I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical point of an elementary embedding $j: V \to M$ such that $M$ is closed under sequences of length \sup\set{j(f)(\kappa) \st f: \kappa \to \kappa}. Some of the other cardinals analyzed include the super-high-jump cardinals, almost-high-jump cardinals, Shelah-for-supercompactness cardinals, Woodin-for-supercompactness cardinals, \Vopenka\ cardinals, hypercompact cardinals, and enhanced supercompact cardinals. I organize these cardinals in terms of consistency strength and implicational strength. I also analyze the superstrong cardinals, which are weaker than supercompact cardinals but are related to high-jump cardinals. Two of my most important results are as follows. \begin{itemize} \item \Vopenka\ cardinals are the same as Woodin-for-supercompactness cardinals. \item There are no excessively hypercompact cardinals. \end{itemize} Furthermore, I prove some results relating high-jump cardinals to forcing, as well as analyzing Laver functions for super-high-jump cardinals. \keywords{high-jump cardinals \and \Vopenka\ cardinals \and Woodin-for-supercompactness cardinals \and hypercompact cardinals \and forcing and large cardinals \and Laver functions

Filter-linkedness and its effect on preservation of cardinal characteristics

We introduce the property ``$F$-linked'' of subsets of posets for a given free filter $F$ on the natural numbers, and define the properties ``$\mu$-$F$-linked'' and ``$\theta$-$F$-Knaster'' for posets in a natural way. We show that $\theta$-$F$-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. Concerning iterations of such posets, we develop a general technique to construct $\theta$-$\mathrm{Fr}$-Knaster posets (where $\mathrm{Fr}$ is the Frechet ideal) via matrix iterations of ${<}\theta$-ultrafilter-linked posets (restricted to some level of the matrix). This is applied to prove consistency results about Cicho\'n's diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal invariants associated with it are pairwise different. At the end, we show that three strongly compact cardinals are enough to force that Cicho\'n's diagram can be separated into $10$ different values.Comment: 30 pages, 7 figure

Lower consistency bounds for mutual stationarity with divergent uncountable cofinalities

We prove that the upper bounds for the consistency strength of certain instances of mutual stationarity considered by Liu-Shelah~\cite{MR1469093} are close to optimal. We also consider some related and, as it turns out, stronger properties

A small ultrafilter number at smaller cardinals

It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a supercompact cardinal that there is a uniform ultrafilter on ${\aleph}_{\omega+1}$ which is generated by fewer than ${2}^{{\aleph}_{\omega+1}}$ sets.Comment: 8 pages. Submitte

Borel's Conjecture in Topological Groups

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $\kappa$, let {\sf BC}$_{\kappa}$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_{\kappa}$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: (1)If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\sf BC}_{\aleph_1}$. (2)If it is consistent that ${\sf BC}_{\aleph_1}$ holds, then it is consistent that there is an inaccessible cardinal. (3)If it is consistent that there is a 1-inaccessible cardinal with $\omega$ inaccessible cardinals above it, then $\neg{\sf BC}_{\aleph_{\omega}} \, +\, (\forall n<\omega){\sf BC}_{\aleph_n}$ is consistent. (4)If it is consistent that there is a 2-huge cardinal, then it is consistent that ${\sf BC}_{\aleph_{\omega}}$. (5)If it is consistent that there is a 3-huge cardinal, then it is consistent that ${\sf BC}_{\kappa}$ holds for a proper class of cardinals $\kappa$ of countable cofinality.Comment: 15 page

Invariants Related to the Tree Property

We consider global analogues of model-theoretic tree properties. The main objects of study are the invariants related to Shelah's tree property $\kappa_{\text{cdt}}(T)$, $\kappa_{\text{sct}}(T)$, and $\kappa_{\text{inp}}(T)$ and the relations that obtain between them. From strong colorings, we construct theories $T$ with $\kappa_{\text{cdt}}(T) > \kappa_{\text{sct}}(T) + \kappa_{\text{inp}}(T)$. We show that these invariants have distinct structural consequences, by investigating the decay of saturation in ultrapowers of models of $T$, where $T$ is some theory with $\kappa_{\text{cdt}}(T)$, $\kappa_{\text{sct}}(T)$, or $\kappa_{\text{inp}}(T)$ large and bounded. This answers some questions of Shelah

On Ciesielski's problems

We discuss some problems posed by Ciesielski. For example we show that, consistently, d_c is a singular cardinal and e_c<d_c. Next we prove that the Martin Axiom for sigma --centered forcing notions implies that for every function f:R^2 ---> R there are functions g_n,h_n:R ---> R, n< omega, such that f(x,y)= sum_{n=0}^{infty} g_n(x)h_n(y). Finally, we deal with countably continuous functions and we show that in the Cohen model they are exactly the functions f with the property that (for all U in [R]^{aleph_1})(exists U^* in [U]^{aleph_1}) (f restriction U^* is continuous)

Knaster and friends I: Closed colorings and precalibers

The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of $\kappa$-cc posets whose squares are not $\kappa$-cc were constructed by Laver, Galvin, Roitman and Fleissner. Later, $\mathsf{ZFC}$ examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which $\kappa = \aleph_2$, was resolved by Shelah in 1997. In this work, we obtain analogous results regarding the infinite productivity of strong chain conditions, such as the Knaster property. Among other results, for any successor cardinal $\kappa$, we produce a $\mathsf{ZFC}$ example of a poset with precaliber $\kappa$ whose $\omega^{\mathrm{th}}$ power is not $\kappa$-cc. To do so, we carry out a systematic study of colorings satisfying a strong unboundedness condition. We prove a number of results indicating circumstances under which such colorings exist, in particular focusing on cases in which these colorings are moreover closed