On critical cardinalities related to QQ-sets

In this note we collect some known information and prove new results about the small uncountable cardinal $\mathfrak q_0$. The cardinal $\mathfrak q_0$ is defined as the smallest cardinality $|A|$ of a subset $A\subset \mathbb R$ which is not a $Q$-set (a subspace $A\subset\mathbb R$ is called a $Q$-set if each subset $B\subset A$ is of type $F_\sigma$ in $A$). We present a simple proof of a folklore fact that $\mathfrak p\le\mathfrak q_0\le\min\{\mathfrak b,\mathrm{non}(\mathcal N),\log(\mathfrak c^+)\}$, and also establish the consistency of a number of strict inequalities between the cardinal $\mathfrak q_0$ and other standard small uncountable cardinals. This is done by combining some known forcing results. A new result of the paper is the consistency of $\mathfrak{p} < \mathfrak{lr} < \mathfrak{q}_0$, where $\mathfrak{lr}$ denotes the linear refinement number. Another new result is the upper bound $\mathfrak q_0\le\mathrm{non}(\mathcal I)$ holding for any $\mathfrak q_0$-flexible cccc $\sigma$-ideal $\mathcal I$ on $\mathbb R$.Comment: 8 page

The tree property at first and double successors of singular cardinals with an arbitrary gap

Let $\mathrm{cof}(\mu)=\mu$ and $\kappa$ be a supercompact cardinal with $\mu<\kappa$. Assume that there is an increasing and continuous sequence of cardinals $\langle\kappa_\xi\mid \xi<\mu\rangle$ with $\kappa_0:=\kappa$ and such that, for each $\xi<\mu$, $\kappa_{\xi+1}$ is supercompact. Besides, assume that $\lambda$ is a weakly compact cardinal with $\sup_{\xi<\mu}\kappa_\xi<\lambda$. Let $\Theta\geq\lambda$ be a cardinal with $\mathrm{cof}(\Theta)>\kappa$. Assuming the $\mathrm{GCH}_{\geq\kappa}$, we construct a generic extension where $\kappa$ is strong limit, $\mathrm{cof}(\kappa)=\mu$, $2^\kappa= \Theta$ and both $\mathrm{TP}(\kappa^+)$ and $\mathrm{TP}(\kappa^{++})$ hold. Further, in this model there is a very good and a bad scale at $\kappa$. This generalizes the main results of [Sin16a] and [FHS18]

Partition properties for simply definable colourings

We study partition properties for uncountable regular cardinals that arise by restricting partition properties defining large cardinal notions to classes of simply definable colourings. We show that both large cardinal assumptions and forcing axioms imply that there is a homogeneous closed unbounded subset of $\omega_1$ for every colouring of the finite sets of countable ordinals that is definable by a $\Sigma_1$-formula that only uses the cardinal $\omega_1$ and real numbers as parameters. Moreover, it is shown that certain large cardinal properties cause analogous partition properties to hold at the given large cardinal and these implications yield natural examples of inaccessible cardinals that possess strong partition properties for $\Sigma_1$-definable colourings and are not weakly compact. In contrast, we show that $\Sigma_1$-definability behaves fundamentally different at $\omega_2$ by showing that various large cardinal assumptions and \emph{Martin's Maximum} are compatible with the existence of a colouring of pairs of elements of $\omega_2$ that is definable by a $\Sigma_1$-formula with parameter $\omega_2$ and has no uncountable homogeneous set. Our results will also allow us to derive tight bounds for the consistency strengths of various partition properties for definable colourings. Finally, we use the developed theory to study the question whether certain homeomorphisms that witness failures of weak compactness at small cardinals can be simply definable.Comment: 28 page

Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata

We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.Comment: 13 page

Wide gaps with short extenders

Let kappa be the limit of (1) if each kappa_n carries an extender of the length of the first Mahlo above kappa_n, then for every ld above kappa there is a generic extension with power of kappa above ld. (2) if each kappa_n carries an extender of the length of the first fixed point of the aleph function above kappa_n of order n then for every ld between kappa and the first inaccessible above kappa there is a generic extension satisfying 2^kappa>ld.Comment: 15 pages, LaTe

The Ultrapower Axiom and the equivalence between strong compactness and supercompactness

The relationship between the large cardinal notions of strong compactness and supercompactness cannot be determined under the standard ZFC axioms of set theory. Under a hypothesis called the Ultrapower Axiom, we prove that the notions are equivalent except for a class of counterexamples identified by Menas. This is evidence that strongly compact and supercompact cardinals are equiconsistent

On a conjecture of Tarski on products of cardinals

We look at an old conjecture of A. Tarski on cardinal arithmetic and show that if a counterexample exists, then there exists one of length omega_1 + omega

Pointwise Definable Models of Set Theory

A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V=HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Godel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.Comment: 23 page

Strongly almost disjoint families, II

The relations M(kappa,lambda,mu)->B [resp. B(sigma)] meaning that if A subset [kappa]^lambda with |A|=kappa is mu-almost disjoint then A has property B [resp. has a sigma-transversal] had been introduced and studied under GCH by Erdos and Hajnal in 1961. Our two main results here say the following: Assume GCH and rho be any regular cardinal with a supercompact [resp. 2-huge] cardinal above rho. Then there is a rho-closed forcing P such that, in V^P, we have both GCH and M(rho^{(+rho+1)},rho^+,rho) not-> B [resp. M(rho^{(+rho+1)},lambda,rho) not-> B(rho^+) for all lambda =< rho^{(+rho+1)}]

Simultaneous stationary reflection and square sequences

We investigate the relationship between weak square principles and simultaneous reflection of stationary sets