Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete. Further we observe that if $V$ is a model of ZF + $DC_{\kappa}$, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-strategically closed and $\mathcal{F}$ is $\kappa$-complete.Comment: Revised versio

Generalizing random real forcing for inaccessible cardinals

The two parallel concepts of "small" sets of the real line are meagre sets and null sets. Those are equivalent to Cohen forcing and Random real forcing for aleph_0^aleph_0; in spite of this similarity, the Cohen forcing and Random Real Forcing have very different shapes. One of these differences is in the fact that the Cohen forcing has an easy natural generalization for lambda 2 while lambda greater than aleph 0, corresponding to an extension for the meagre sets, while the Random real forcing didn't seem to have a natural generalization, as Lebesgue measure doesn't have a generalization for space 2 lambda while lambda greater than aleph 0. In work [1], Shelah found a forcing resembling the properties of Random Real Forcing for 2 lambda while lambda is a weakly compact cardinal. Here we describe, with additional assumptions, such a forcing for 2 lambda while lambda is an Inaccessible Cardinal; this forcing is less than lambda-complete and satisfies the lambda^+-c.c hence preserves cardinals and cofinalities, however unlike Cohen forcing, does not add an undominated real

Indestructibility of Vopenka's Principle

We show that Vopenka's Principle and Vopenka cardinals are indestructible under reverse Easton forcing iterations of increasingly directed-closed partial orders, without the need for any preparatory forcing. As a consequence, we are able to prove the relative consistency of these large cardinal axioms with a variety of statements known to be independent of ZFC, such as the generalised continuum hypothesis, the existence of a definable well-order of the universe, and the existence of morasses at many cardinals.Comment: 15 pages, submitted to Israel Journal of Mathematic

L-like Combinatorial Principles and Level by Level Equivalence

We force and construct a model in which GCH and level by level equivalence between strong compactness and supercompactness hold, along with certain additional “L-like ” combinatorial principles. In particular, this model satisfies the following properties: 1. ♦δ holds for every successor and Mahlo cardinal δ. 2. There is a stationary subset S of the least supercompact cardinal κ0 such that for every δ ∈ S, ¤δ holds and δ carries a gap 1 morass. 3. A weak version of ¤δ holds for every infinite cardinal δ. 4. There is a locally defined well-ordering of the universe W, i.e., for all κ ≥ ℵ2 a regular cardinal, W H(κ+) is definable over the structure 〈H(κ+),∈ 〉 by a parameter free formula. ∗2000 Mathematics Subject Classifications: 03E35, 03E55. †Keywords: Supercompact cardinal, strongly compact cardinal, strong cardinal, level by level equivalence between strong compactness and supercompactness, diamond, square, morass, locally defined well-ordering. ‡The author’s research was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive grants. §The author wishes to thank the referee for helpful comments, suggestions, and corrections which have been incorporated into the current version of the paper. 1 The model constructed amalgamates and synthesizes results due to the author, the author and Cummings, and Aspero ́ and Sy Friedman. It has no restrictions on the structure of its class of supercompact cardinals and may be considered as part of Friedman’s “outer model programme”.