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LOCAL CLUB CONDENSATION AND L-LIKENESS

Published online by Cambridge University Press:  22 December 2015

PETER HOLY ,
PHILIP WELCH  and
LIUZHEN WU
PETER HOLY
Affiliation:
MATHEMATISCHES INSTITUT RHEINISCHE FRIEDRICH-WILHELMS-UNIVERSITÄT BONN ENDENICHER ALLEE 60 53115 BONN GERMANYE-mail: pholy@math.uni-bonn.de
PHILIP WELCH
Affiliation:
UNIVERSITY OF BRISTOL DEPARTMENT OF MATHEMATICS UNIVERSITY WALK BRISTOL BS8 1TW, UKE-mail: p.welch@bristol.ac.uk
LIUZHEN WU
Affiliation:
INSTITUTE OF MATHEMATICS CHINESE ACADEMY OF SCIENCES EAST ZHONG GUAN CUN ROAD 55 100190 BEIJING CHINAE-mail: lzwu@math.ac.cn
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Abstract

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.

Keywords

03E3503E55CondensationL-like modelSquareChang’s ConjectureForcingErdős cardinalInterleaving Structures
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 4 , December 2015 , pp. 1361 - 1378
Copyright
Copyright © The Association for Symbolic Logic 2015 

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References

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