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On Namba Forcing and Minimal Collapses

Maxwell Levine

TL;DR

The paper tackles the problem of controlling cardinal collapses in Namba-type forcing without assuming the Continuum Hypothesis. It introduces a Laver Ideal Property and a tall augmented Namba forcing to obtain refined $|\lambda|=\kappa$-extensions, including a $|\aleph_2^V|=\aleph_1$-minimal extension not an $|\aleph_3^V|=\aleph_1$-extension. The main method combines a sweeping argument with a fusion technique to produce countably closed, minimal extensions under a measurable cardinal. These results demonstrate greater flexibility in constructing minimal extensions and clarify how large cardinals influence the interaction between minimality and cardinal collapse in tree-like forcings.

Abstract

We build on a 1990 paper of Bukovsky and Coplakova-Hartova. First, we remove the hypothesis of $\textsf{CH}$ from one of their minimality results. Then, using a measurable cardinal, we show that there is a $|\aleph_2^V|=\aleph_1$-minimal extension that is not a $|\aleph_3^V|=\aleph_1$-extension, answering the first of their questions.

On Namba Forcing and Minimal Collapses

TL;DR

The paper tackles the problem of controlling cardinal collapses in Namba-type forcing without assuming the Continuum Hypothesis. It introduces a Laver Ideal Property and a tall augmented Namba forcing to obtain refined -extensions, including a -minimal extension not an -extension. The main method combines a sweeping argument with a fusion technique to produce countably closed, minimal extensions under a measurable cardinal. These results demonstrate greater flexibility in constructing minimal extensions and clarify how large cardinals influence the interaction between minimality and cardinal collapse in tree-like forcings.

Abstract

We build on a 1990 paper of Bukovsky and Coplakova-Hartova. First, we remove the hypothesis of from one of their minimality results. Then, using a measurable cardinal, we show that there is a -minimal extension that is not a -extension, answering the first of their questions.
Paper Structure (6 sections, 7 theorems, 4 equations)

This paper contains 6 sections, 7 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Definitions and Notation
  3. Results
  4. Minimality without the Continuum Hypothesis
  5. Developing a Version of Higher Namba Forcing
  6. Remaining Questions

Key Result

lemma 4

If $G$ is $\mathbb{P}_{\textup{CNF}}$-generic over $V$ and suppose $f \in V[G]$ is an unbounded function $\omega \to \theta$ where $\mathop{\mathrm{cf}}\nolimits^V(\theta) \ge \aleph_2$. Then $V[f]=V[G]$.

Theorems & Definitions (21)

  • definition 1
  • definition 2
  • definition 3
  • lemma 4
  • proof
  • theorem 5
  • proof
  • definition 6: Laver
  • definition 8
  • proposition 9
  • ...and 11 more