On the Hamkins approximation property

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Abstract

We give a short proof of a lemma which generalizes both the main lemma from the original construction in the author’s thesis of a model with no ω2-Aronszajn trees, and also the “Key Lemma” in Hamkins’ gap forcing theorems. The new lemma directly yields Hamkins’ newer lemma stating that certain forcing notions have the approximation property.

Keywords

Forcing
Gap forcing
Approximation property

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This paper is dedicated to Jim Baumgartner. My relations with him as a friend and colleague date back to the time I was working on my thesis, to which Jim made valuable contributions and to which I return in this note.