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CLASS FORCING, THE FORCING THEOREM AND BOOLEAN COMPLETIONS

Published online by Cambridge University Press:  01 December 2016

PETER HOLY ,
REGULA KRAPF ,
PHILIPP LÜCKE ,
ANA NJEGOMIR  and
PHILIPP SCHLICHT
PETER HOLY
Affiliation:
MATH. INSTITUT UNIVERSITÄT BONN ENDENICHER ALLEE 60, 53115 BONN, GERMANY E-mail: pholy@math.uni-bonn.de
REGULA KRAPF
Affiliation:
MATH. INSTITUT UNIVERSITÄT BONN ENDENICHER ALLEE 60, 53115 BONN, GERMANY E-mail: krapf@math.uni-bonn.de
PHILIPP LÜCKE
Affiliation:
MATH. INSTITUT UNIVERSITÄT BONN ENDENICHER ALLEE 60, 53115 BONN, GERMANY E-mail: pluecke@math.uni-bonn.de
ANA NJEGOMIR
Affiliation:
MATH. INSTITUT UNIVERSITÄT BONN ENDENICHER ALLEE 60, 53115 BONN, GERMANY E-mail: njegomir@math.uni-bonn.de
PHILIPP SCHLICHT
Affiliation:
MATH. INSTITUT UNIVERSITÄT BONN ENDENICHER ALLEE 60, 53115 BONN, GERMANY INSTITUT FÜR MATHEMATISCHE LOGIK UND GRUNDLAGENFORSCHUNG UNIVERSITÄT MÜNSTER EINSTEINSTR. 62, 48149 MÜNSTER, GERMANY E-mail: schlicht@math.uni-bonn.de
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Abstract

The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing.

In addition to these negative results, we show that the forcing theorem is equivalent to the existence of a (certain kind of) Boolean completion, and we introduce a weak combinatorial property (approachability by projections) that implies the forcing theorem to hold. Finally, we show that unlike for set forcing, Boolean completions need not be unique for class forcing.

Keywords

03E4003E7003E99class forcingforcing theoremBoolean completions

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 4 , December 2016 , pp. 1500 - 1530
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

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