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The ZFC analogue of the Halpern-Levy theorem

Nedeljko Stefanović

TL;DR

Addresses how ZFC can yield Halpern-Läuchli results by replacing metamathematical arguments with ZFC theorems. Introduces a Polish space lemma and a condensational subtree framework to obtain HL-type colorings. Shows robustness across Cohen forcing and random forcing, and connects to coloring principles DDF and PG, broadening ZFC-based approaches to HL.

Abstract

Here we present ZFC theorems yielding the Halpern-L\a"uchli theorem and avoiding metamathematical notions in their formulations.

The ZFC analogue of the Halpern-Levy theorem

TL;DR

Addresses how ZFC can yield Halpern-Läuchli results by replacing metamathematical arguments with ZFC theorems. Introduces a Polish space lemma and a condensational subtree framework to obtain HL-type colorings. Shows robustness across Cohen forcing and random forcing, and connects to coloring principles DDF and PG, broadening ZFC-based approaches to HL.

Abstract

Here we present ZFC theorems yielding the Halpern-L\a"uchli theorem and avoiding metamathematical notions in their formulations.
Paper Structure (4 sections, 11 theorems, 22 equations)

This paper contains 4 sections, 11 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Cohen's symmetric model and its generalizations
  3. Properties of Cohen's symmetric model
  4. The results

Key Result

Theorem 1

audrito ( Laver's Theorem) If a transitive model $M$ satisfies a suitable finite fragment of the theory $\mathrm{ZFC}$ and if $P\in M$ is some atomless partial ordering, and $G$ is some $P$-generic filter over $M$, then $M$ is a class in the model $M[G]$, definable with a parameter in $M$.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Lemma 1
  • Theorem 9
  • ...and 1 more