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Switch to: Citations
References in:
The lottery preparation
Annals of Pure and Applied Logic 101 (2-3):103-146 (2000)
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The Lévy-Solovay Theorem [8] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on. |
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Laver [L] and others [G-S] have shown how to make the supercompactness or strongness of κ indestructible by a wide class of forcing notions. We show, alternatively, how to make these properties fragile. Specifically, we prove that it is relatively consistent that any forcing which preserves $\kappa^{<\kappa}$ and κ+, but not P(κ), destroys the measurability of κ, even if κ is initially supercompact, strong, or if I1(κ) holds. Obtained as an application of some general lifting theorems, this result is an (...) |
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The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of what I call (...) |
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Applying the seed concept to Prikry tree forcing P μ , I investigate how well P μ preserves the maximality property of ordinary Prikry forcing and prove that P μ Prikry sequences are maximal exactly when μ admits no non-canonical seeds via a finite iteration. In particular, I conclude that if μ is a strongly normal supercompactness measure, then P μ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. Hugh Woodin's. |
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A model in which strongness ofκ is indestructible under κ+ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of {λ δ |2 δ <λ}. |
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In this paper we introduce the Wholeness Axiom , which asserts that there is a nontrivial elementary embedding from V to itself. We formalize the axiom in the language {∈, j } , adding to the usual axioms of ZFC all instances of Separation, but no instance of Replacement, for j -formulas, as well as axioms that ensure that j is a nontrivial elementary embedding from the universe to itself. We show that WA has consistency strength strictly between I 3 (...) |
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Versions of Laver sequences are known to exist for supercompact and strong cardinals. Assuming very strong axioms of infinity, Laver sequences can be constructed for virtually any globally defined large cardinal not weaker than a strong cardinal; indeed, under strong hypotheses, Laver sequences can be constructed for virtually any regular class of embeddings. We show here that if there is a regular class of embeddings with critical point κ, and there is an inaccessible above κ, then it is consistent for (...) |
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We show the consistency, relative to a supercompact cardinal, of the least measurable cardinal being both strongly compact and fully Laver indestructible. We also show the consistency, relative to a supercompact cardinal, of the least strongly compact cardinal being somewhat supercompact yet not completely supercompact and having both its strong compactness and degree of supercompactness fully Laver indestructible. |
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We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V “ZFC + Ω is the least inaccessible limit of measurable limits of supercompact cardinals + ƒ : Ω → 2 is a function”, then there is a partial ordering P V so that for , There is a proper class of compact cardinals + If (...) |
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