Related papers: Clarifying ordinals

If T is an iteration tree on K and F is a countably certified extender that coheres with the final model of T, then F is on the extender sequence of the final model of T. Several applications of maximality are proved, including: o K…

Let $M$ be a transitive model of $ZFC$ and let ${\bf B}$ be a $M$-complete Boolean algebra in $M.$ (In general a proper class.) We define a generalized notion of forcing with such Boolean algebras, $^*$forcing. (A $^*$ forcing extension of…

We present a forcing for blowing up 2^lambda and making ``many positive polarized partition relations'' (in a sense made precise in (c) of our main theorem) hold in the interval [lambda, 2^lambda]. This generalizes results of [276], Section…

In this paper we extend the approach of M. Cavaleri to effective amenability to the class of computably enumerable groups, i.e. in particular we do not assume that groups are finitely generated. In the case of computable groups we also…

It is shown that the existence of a measurable cardinal is equiconsistent to a model of ZFC in which there is no ordinal-definable, stationary, costationary subset of $\omega_1$

We characterize pairs of orthogonal countable ordinals. Two ordinals $\alpha$ and $\beta$ are orthogonal if there are two linear orders $A$ and $B$ on the same set $V$ with order types $\alpha$ and $\beta$ respectively such that the only…

Measurability with respect to ideals is tightly connected with absoluteness principles for certain forcing notions. We study a uniformization principle that postulates the existence of a uniformizing function on a large set, relative to a…

We define and investigate HC-forcing invariant formulas of set theory, whose interpretations in the hereditarily countable sets are well behaved under forcing extensions. This leads naturally to a notion of cardinality ||Phi|| for sentences…

In this paper we investigate more characterizations and applications of $\delta$-strongly compact cardinals. We show that, for a cardinal $\kappa$ the following are equivalent: (1) $\kappa$ is $\delta$-strongly compact, (2) For every…

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals…

For an exotic locally compact Hausdorff space $L$, constructed under the assumption of the Ostaszewski's $\clubsuit$-principle, and a countable ordinal space $\alpha$, we prove that all operators defined on $C_0(\alpha\times L)$ are as…

Hellsten \cite{MR2026390} proved that when $\kappa$ is $\Pi^1_n$-indescribable, the \emph{$n$-club} subsets of $\kappa$ provide a filter base for the $\Pi^1_n$-indescribability ideal, and hence can also be used to give a characterization of…

In $\mathsf{ZFC}$, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation $E \in L(\mathbb{R})$ on $\mathbb{R}$ with all $\mathbf{\Delta}_1^1$ classes and every $\sigma$-ideal…

We call a subset of an ordinal $\lambda$ recognizable if it is the unique subset $x$ of $\lambda$ for which some Turing machine with ordinal time and tape, which halts for all subsets of $\lambda$ as input, halts with the final state $0$.…

We give characterizations for the (in ZFC unprovable) sentences ``Every $\Sb{1}{2}$--set is measurable" and ``Every $\Db{1}{2}$--set is measurable" for various notions of measurability derived from well--known forcing partial orderings.

In various models of set theory, we consider covering Aleph_1 x Aleph_1 rectangles by countably many smooth curves, and we study differentiable isomorphisms between Aleph_1-dense sets of reals.

We show a number of undecidable assertions concerning countably compact spaces hold under PFA(S)[S]. We also show the consistency without large cardinals of "every locally compact, perfectly normal space is paracompact".

We mainly investigate model of set theory with restricted choice, e.g., ZF + DC + "the family of countable subsets of lambda is well ordered for every lambda" (really local version for a given lambda). In this frame much of pcf theory can…

We prove that if there are $\mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $\kappa$ such that $\kappa^\omega=\kappa$, there exists a group topology on the free Abelian group of cardinality $\kappa$…

We exhibit a way of "forcing a functional to be an effective operation" for arbitrary partial combinatory algebras (pcas). This gives a method of defining new pcas from old ones for some fixed functional, where the new partial functions can…