A large Pi-1-2 set absolute for set forcing

Sy D. Friedman

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arXiv · math/9212202

A large Pi-1-2 set absolute for set forcing

Sy D. Friedman

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Abstract

Let k be a definable L-cardinal. Then there is a set of reals X, class-generic over L, such that L(X) and L have the same cardinals, X has size k in L(X) and some pi-1-2 formula defines X in all set-generic extensions of L(X). Two corollaries, both assuming the consistency of an inaccessible: It is consistent for the Perfect Set Property to hold for boldface sigma-1-2 sets, yet fail for some lightface pi-1-2 set. It is consistent that the Perfect Set Property holds for boldface sigma-1-2 sets yet some lightface pi-1-2 wellordering of some set of reals has length aleph-1000.

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BibTeX RIS

Sy D. Friedman. 1992-12-02. A large Pi-1-2 set absolute for set forcing. https://arxiv.org/abs/math/9212202

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