On the Hamkins approximation property

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’s gap forcing theorems. The new lemma directly yields Hamkins’s newer lemma stating that certain forcing notions have the approximation property. According to Hamkins [2], a partial ordering P satisfies the δ-approximation property if, whenever A ∈ V P is a subset of an ordinal μ in V P such that A ∩ x ∈ V for each x ∈ ([μ]) , we have A ∈ V . In [2, Lemma 13] he proves the following lemma for the case when δ is a successor cardinal: Lemma 1. Suppose that δ is a regular cardinal, P ∗ Q̇ is a forcing in which P is nontrivial and |P | < δ, and P Q̇ is <δ-strategically closed. Then P ∗ Q̇ has the δ-approximation property. This is a generalization of the “Key Lemma” of Hamkins’s gap forcing theorems [1, 2]. Hamkins proves Lemma 1 from these key lemmas, which in turn are proved using a tree argument which is quite similar to the proof of the main lemma in the original construction [3] of a model with no ω1-Aronszajn tree from a weakly compact cardinal κ. That model is constructed using a forcing which can be described in modern terms as a forcing iteration 〈 (Pν , Q̇ν) : ν < κ 〉 of length κ. If ν < κ is a regular cardinal then Qν is the forcing to add a Cohen real, and Q̇ν+1 is a name for the forcing to collapse ν onto ω1; the forcing Qν is trivial for all other ordinals ν < κ. The iteration uses finite support for the Cohen reals and countable support for the collapses. The main lemma of [3] states: Lemma 2. Suppose that G ⊂ Pκ is generic, that λ < κ is a regular cardinal, and b is a branch in V [G] of a tree T ∈ V [Gλ] of height λ, where Gλ is the restriction of G to Pλ, the first λ stages of the iteration. Then b ∈ V [Gλ]. ∗Work on this paper was partly supported by grant number DMS-0400954 from the National Science Foundation. 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.