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”.

Inner models with large cardinal features usually obtained by forcing

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal \kappa for which 2^\kappa=\kappa^+, another for which 2^\kappa=\kappa^++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal \kappa, such that H_{\kappa^+}^V\subseteq HOD^W. Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH+V=HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit \delta of <\delta-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results

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

A new test of conservation laws and Lorentz invariance in relativistic gravity

General relativity predicts that energy and momentum conservation laws hold and that preferred frames do not exist. The parametrised post-Newtonian formalism (PPN) phenomenologically quantifies possible deviations from general relativity. The PPN parameter alpha_3 (which identically vanishes in general relativity) plays a dual role in that it is associated both with a violation of the momentum conservation law, and with the existence of a preferred frame. By considering the effects of alpha_3 neq 0 in certain binary pulsar systems, it is shown that alpha_3 < 2.2 x 10^-20 (90% CL). This limit improves on previous results by several orders of magnitude, and shows that pulsar tests of alpha_3 rank (together with Hughes-Drever-type tests of local Lorentz invariance) among the most precise null experiments of physics.Comment: Submitted to Classical Quantum Gravity, LaTeX, requires ioplppt.sty, no figure