Abstract

We prove that if the Mathias forcing is followed by a forcing with the Laver Property, then any $$\mathsf {V}$$V-$$\mathsf {q}$$q-point is isomorphic via a ground model bijection to the canonical $$\mathsf {V}$$V-Ramsey ultrafilter added by the Mathias real. This improves a result of Shelah and Spinas (Trans AMS 325:2023---2047, 1999).

Highlights

  • We prove: Theorem Suppose that r is a Mathias real over V, P ∈ V[r ] is a poset that has the Laver property in V[r ], GP ⊆ P is a generic filter over V[r ], and x ∈ [ω]ω ∩ V[r ][GP ] is such that x ∩ V is a V-ultrafilter

  • Lemma 4 Suppose that Q Phas the Laver property, and that

  • This ends the proof of Technical Lemma

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Summary

Introduction

The proof in [2] is somewhat demanding.1 We prove: Theorem Suppose that r is a Mathias real over V, P ∈ V[r ] is a poset that has the Laver property in V[r ], GP ⊆ P is a generic filter over V[r ], and x ∈ [ω]ω ∩ V[r ][GP ] is such that x ∩ V is a V-ultrafilter. V-Ramsey and V[r ][ rξ ξ

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