Related papers: Prikry On Extenders, Revisited
We study which $\kappa$-distributive forcing notions of size $\kappa$ can be embedded into tree Prikry forcing notions with $\kappa$-complete ultrafilters under various large cardinal assumptions. An alternative formulation -- can the…
Assuming that there is no inner model with a strong cardinal, the following is shown: any subset of \omega_1 can be made \Delta^1_3 (in the codes) by a reasonable set-forcing; there is a reasonable set-generic extension with a \Delta^1_3…
Large cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained that way (`Unfoldable cardinals') behave as…
Continuing \cite{GitJir22}, we develop a version of Extender-based Magidor-Radin forcing where there are no extenders on the top ordinal. As an application, we provide another approach to obtain a failure of SCH on a club subset of an…
Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles…
Suppose that kappa is a singular cardinal of cofinality omega and GCH holds. Assume that for every n<omega the set of alphas with o(alpha)>= alpha^{+n} is unbounded in kappa.Then there is a cardinal preserving extension satisfying…
We give an application of our extender based Radin forcing to cardinal arithmetic. Using a preparation forcing and interleaving of Cohen and Levy forcings in the normal Radin sequence we get a model with a power function having a fixed…
In this paper, we answer a question asked in "A minimal Prikry type forcing for singularizing a measurable cardinal" regarding a Mathias criteria for Tree-Prikry forcing. Also we will investigate Prikry forcing using various filters. For…
I survey an array of topics in set theory in the context of a novel class of forcing notions: subcomplete forcing. Subcompleteness was originally defined by Ronald Jensen. I have attempted to make the subject somewhat more approachable to…
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…
We develop a toolbox for forcing over arbitrary models of set theory without the axiom of choice. In particular, we introduce a variant of the countable chain condition and prove an iteration theorem that applies to many classical forcings…
We study methods to obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first force over a model with a supercompact cardinal $\theta>\kappa$ to get the consistency of the forcing axiom for $\kappa$-strongly…
We study the Magidor iteration of Prikry forcings below a measurable limit of measurables $ \kappa $. We first characterize all the normal measures $ \kappa $ carries in the generic extension, building on and extending the main result of…
We present two ways in which the model $L({\mathbb R})$ is canonical assuming the existence of large cardinals. We show that the theory of this model, with {\em ordinal} parameters, cannot be changed by small forcing; we show further that a…
We show: There are pairs of universes V_1 subseteq V_2 and there is a notion of forcing P in V_1 such that the change mentioned in the title occurs when going from V_1[G] to V_2[G] for a P-generic filter G over V_2. We use forcing…
We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular…
We continue the work done by the authors and before that by the second author, Kanovei and koepke. We prove that for every set of ordinals $A$ in a Magidor-Radin generic extension using a coherent sequence such that…
Given a Fra\"{i}ss\'{e} class $\mathcal{K}$ and an infinite cardinal $\kappa,$ we define a forcing notion which adds a structure of size $\kappa$ using elements of $\mathcal{K}$, which extends the Fra\"{i}ss\'{e} construction in the case…
We define a version of Gitik-Sharon diagonal Prikry forcing using a strongly compact cardinal, and prove its basic properties.
Cicho\'n's diagram describes the connections between combinatorial notions related to measure, category, and compactness of sets of irrational numbers. In the second part of the 2010's, Goldstern, Kellner and Shelah constructed a forcing…