CLASS FORCING, THE FORCING THEOREM AND BOOLEAN COMPLETIONS. (1st December 2016)

• Record Type: Journal Article
• Authors: HOLY, PETER; KRAPF, REGULA; LÜCKE, PHILIPP; NJEGOMIR, ANA; SCHLICHT, PHILIPP
• Is Part Of: Journal of symbolic logic. Volume 81:Number 4(2016)
• Page Start: 1500
• Page End: 1530
• Publication Date: 2016-12-01
• Subjects: 03E40, -- 03E70, -- 03E99; class forcing, -- forcing theorem, -- Boolean completions; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2016.4
• Languages: English
• ISSNs: 0022-4812

Abstract

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.