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On fresh sets in iterations of Prikry type forcing notions

Moti Gitik, Eyal Kaplan

Abstract

We examine the existence (and mostly non-existence) of fresh sets in commonly used iterations of Prikry type forcing notions. Results of [4] are generalized. As an application, a question of a referee of [9] is answered. In addition stationary sets preservation is addressed.

On fresh sets in iterations of Prikry type forcing notions

Abstract

We examine the existence (and mostly non-existence) of fresh sets in commonly used iterations of Prikry type forcing notions. Results of [4] are generalized. As an application, a question of a referee of [9] is answered. In addition stationary sets preservation is addressed.
Paper Structure (5 sections, 23 theorems, 65 equations)

This paper contains 5 sections, 23 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. On preservation of stationarity in Prikry type extensions
  3. Non-stationary support iterations
  4. On fresh sets in the Easton support iterations
  5. No fresh subsets of $\kappa$ in the full support

Key Result

Theorem 2.2

Assume GCH. Let $\kappa$ be an inaccessible cardinal. Let $P = P_{\kappa}$ be as in the introduction. Let $G\subseteq P_{\kappa}$ be generic over $V$. Then $\kappa$ is inaccessible in $V\left[G\right]$, and--

Theorems & Definitions (34)

  • Definition 1.1
  • Remark 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Corollary 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • ...and 24 more