Local Club Condensation and L-Likeness

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]. 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. 1. Condensation and L-likeness 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 Gödels Condensation Lemma) and other Llike principles; we investigate the question of how close to Gödels constructible universe the universe of sets has to be given that it satisfies certain generalized Condensation principles. For definitions of generalized Condensation principles that will be relevant to this paper see Section 2. 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. This was further improved in [4] by showing that Local Club Condensation and Acceptability are simultaneously consistent with the existence of ω-superstrong cardinals. It is generally believed that the fine structural properties of L are necessary to prove that various square principles hold in L. In [14], the third author showed that Strong Condensation for ω2 is consistent with ¬ ω1 from a stationary limit of measurable cardinals, thus giving additional support to this belief. One of the main aims of this paper is to generalize his result to cardinals beyond ω2, replacing Strong 2010 Mathematics Subject Classification. 03E35, 03E55.