Local club condensation and L-likeness

Peter Holy, Philip D Welch, Liuzhen Wu

Research output: Contribution to journalArticle (Academic Journal)peer-review

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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.

Original languageEnglish
Pages (from-to)1361-1378
Number of pages18
JournalJournal of Symbolic Logic
Issue number4
Early online date22 Dec 2015
Publication statusPublished - Dec 2015


  • Chang’s conjecture
  • Condensation
  • Erdős cardinal
  • Forcing
  • Interleaving structures
  • L-like model
  • Square


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