Abstract
AbstractWe present a forcing to obtain a localized version of Local Club Condensation, a generalized Condensation principle introduced by Sy Friedman and the first author in [3] and [5]. This forcing will have properties nicer than the forcings to obtain this localized version that could be derived from the forcings presented in either [3] or [5]. We also strongly simplify the related proofs provided in [3] and [5]. Moreover our forcing will be capable of introducing this localized principle at κ while simultaneously performing collapses to make κ become the successor of any given smaller regular cardinal. This will be particularly useful when κ has large cardinal properties in the ground model. We will apply this to measure how much L-likeness is implied by Local Club Condensation and related principles. We show that Local Club Condensation at κ+ is consistent with ¬☐κ whenever κ is regular and uncountable, generalizing and improving a result of the third author in [14], and that if κ ≥ ω2 is regular, CC(κ+) - Chang’s Conjecture at κ+ - is consistent with Local Club Condensation at κ+, both under suitable large cardinal consistency assumptions.
Highlights
We present a forcing to obtain a localized version of Local Club Condensation, a generalized Condensation principle introduced by Sy Friedman and the first author in [3] and [5]
Besides the presentation of the forcing announced in the abstract, the central theme of this paper is the relationship between generalized Condensation principles and other Llike principles; we investigate the question of how close to Godels constructible universe the universe of sets has to be given that it satisfies certain generalized Condensation principles
In [3], Sy Friedman and the first author showed that Local Club Condensation allows for the existence of very large large cardinals, far beyond those compatible with V = L - namely they showed, by using the method of forcing, that Local Club Condensation is consistent with the existence of ω-superstrong cardinals
Summary
Besides the presentation of the forcing announced in the abstract, the central theme of this paper is the relationship between generalized Condensation principles (i.e. generalizations of consequences of Godels Condensation Lemma) and other Llike principles; we investigate the question of how close to Godels constructible universe the universe of sets has to be given that it satisfies certain generalized Condensation principles. In [3], Sy Friedman and the first author showed that Local Club Condensation allows for the existence of very large large cardinals, far beyond those compatible with V = L - namely they showed, by using the method of forcing, that Local Club Condensation is consistent with the existence of ω-superstrong cardinals. In [14], the third author showed that Strong Condensation for ω2 is consistent with ¬ ω1 from a stationary limit of measurable cardinals, giving additional support to this belief. Condensation, L-like model, Square, Chang’s Conjecture, Forcing, Erdos cardinal, Interleaving Structures. We further investigate weaker square principles, Jonsson cardinals and Chang’s Conjecture style principles, all in the context of generalized Condensation principles Condensation for ω2 by Local Club Condensation at κ for κ ≥ ω21 and reducing the consistency assumption to a 2-Mahlo cardinal. We further investigate weaker square principles, Jonsson cardinals and Chang’s Conjecture style principles, all in the context of generalized Condensation principles
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