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LARGE CARDINALS AND LIGHTFACE DEFINABLE WELL-ORDERS, WITHOUT THE GCH

Published online by Cambridge University Press:  13 March 2015

SY-DAVID FRIEDMAN ,
PETER HOLY  and
PHILIPP LÜCKE
SY-DAVID FRIEDMAN
Affiliation:
UNIVERSITÄT WIEN, KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC, WÄHRINGER STRASSE 25, 1090 WIEN, AUSTRIAE-mail: sdf@logic.univie.ac.at
PETER HOLY
Affiliation:
UNIVERSITY OF BRISTOL, DEPARTMENT OF MATHEMATICS, UNIVERSITY WALK, BRISTOL BS8 1TW, UKE-mail: maxph@bristol.ac.uk
PHILIPP LÜCKE
Affiliation:
RHEINISCHE FRIEDRICH-WILHELMS-UNIVERSITÄT BONN, MATHEMATISCHES INSTITUT, ENDENICHER ALLEE 60, 53115 BONN, GERMANYE-mail: pluecke@math.uni-bonn.de
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Abstract

This paper deals with the question whether the assumption that for every inaccessible cardinal κ there is a well-order of H(κ+) definable over the structure $\langle {\rm{H}}({\kappa ^ + }), \in \rangle$ by a formula without parameters is consistent with the existence of (large) large cardinals and failures of the GCH. We work under the assumption that the SCH holds at every singular fixed point of the ℶ-function and construct a class forcing that adds such a well-order at every inaccessible cardinal and preserves ZFC, all cofinalities, the continuum function, and all supercompact cardinals. Even in the absence of a proper class of inaccessible cardinals, this forcing produces a model of “V = HOD” and can therefore be used to force this axiom while preserving large cardinals and failures of the GCH. As another application, we show that we can start with a model containing an ω-superstrong cardinal κ and use this forcing to build a model in which κ is still ω-superstrong, the GCH fails at κ and there is a well-order of H(κ+) that is definable over H(κ+) without parameters. Finally, we can apply the forcing to answer a question about the definable failure of the GCH at a measurable cardinal.

Keywords

03E3503E4703E55Class forcingDefinable well-ordersFailures of the GCHHODsupercompact cardinalsω-superstrong cardinals
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 1 , March 2015 , pp. 251 - 284
Copyright
Copyright © The Association for Symbolic Logic 2015 

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References

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