| On the Hamkins Approximation Property (2004) | |||||||||||||
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’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] <δ) V, 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 ∗ ˙ Q is a forcing in which P is nontrivial and |P | < δ, and �P ˙ Q is <δ-strategically closed. Then P ∗ ˙ Q 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 | |||||||||||||
Publication details | |||||||||||||
| |||||||||||||