Related papers: Destroying Saturation while Preserving Presaturati…
We present the classical theory of preservation of $\sqsubset$-unbounded families in generic extensions by ccc posets, where $\sqsubset$ is a definable relation of certain type on spaces of real numbers, typically associated with some…
Despite being an established notion in the large cardinal hierarchy, results about Woodin cardinals are sparse in the literature. Here we gather known results about the preservation of Woodin cardinals under certain forcing extensions, as…
Answering a question of Usuba, we show that an extendible cardinal can be preserved by a set forcing that is not a small forcing.
We introduce the class of unshreddable theories, which contains the simple and NIP theories, and prove that such theories have exactly saturated models in singular cardinals, satisfying certain set-theoretic hypotheses. We also give…
In this paper, we obtain the consistency, relative to large cardinals, of the existence of dense ideals on every successor of a regular cardinal simultaneously. Using a consequent transfer principle, we show that in this model there is a…
We answer a question of Woodin by showing that assuming an inaccessible cardinal $\kappa$ which is a limit of ${<}\kappa$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb…
We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $\omega_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The…
We are proving the following: (1) If $\kap$ is a weakly inaccessible then $NS_\kap$ is not $\kap^+$-saturated. (2) If $\kap$ is a weakly inaccessible and $\tet <\kap$ is regular then $NS^\tet_\kap$ is not $\kap^+$-saturated. (3) If $\kap$…
We investigate the problem of when $\leq\lambda$--support iterations of $<\lambda$--complete notions of forcing preserve $\lambda^+$. We isolate a property -- {\em properness over diamonds} -- that implies $\lambda^+$ is preserved and show…
The main result of this paper is a partial answer to [math.LO/9909115, Problem 5.5]: a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give…
We prove a theorem on iterated forcing that can be used for preservation of $\aleph_2$ and $\aleph_1$ in iterations with supports of size $\aleph_1$ of forcings that have amalgamation properties similar to those present in the perfect set…
From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal kappa is fully indestructible by kappa-directed…
Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. We…
We obtain results on the condensation principle called local club condensation. We prove that in extender models an equivalence between the failure of local club condensation and subcompact cardinals holds. This gives a characterization of…
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a…
We give a brief survey on the interplay between forcing axioms and various other non-constructive principles widely used in many fields of abstract mathematics, such as the axiom of choice and Baire's category theorem. First of all we…
We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals…
Assume $\kappa = \kappa^{< \kappa}$ (usually $\aleph_0$ or an inaccessible). We shall deal with iterated forcings preserving ${}^{\kappa>}{\rm Ord}$ and not collapsing cardinals along a linear order $L$. A sufficient condition for this,…
Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang's conjecture, polarized partition relations, and transfer…
A theory $T$ is said to have exact saturation at a singular cardinal $\kappa$ if it has a $\kappa$-saturated model which is not $\kappa^{+}$-saturated. We show, under some set-theoretic assumptions, that any simple theory has exact…