Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Axiom of Choice in the $κ$-Mantle

Andreas Lietz

Abstract

Usuba has asked whether the $κ$-mantle, the intersection of all grounds that extend to $V$ via a forcing of size ${<}κ$, is always a model of ZFC. We give a negative answers by constructing counterexamples where $κ$ is a Mahlo cardinal, $κ=ω_1$ and where $κ$ is the successor of a regular uncountable cardinal.

The Axiom of Choice in the $κ$-Mantle

Abstract

Usuba has asked whether the -mantle, the intersection of all grounds that extend to via a forcing of size , is always a model of ZFC. We give a negative answers by constructing counterexamples where is a Mahlo cardinal, and where is the successor of a regular uncountable cardinal.
Paper Structure (7 sections, 23 theorems, 36 equations)

This paper contains 7 sections, 23 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. The Axiom of Choice May Fail in $\mathbb{M}_{\kappa}$
  4. The case "$\kappa$ is Mahlo"
  5. The $\omega_1$-mantle
  6. The successor of a regular uncountable cardinal case
  7. Conclusion

Key Result

Theorem 2.1

If $\mathrm{ZFC}$ is consistent with the existence of a Mahlo cardinal, then it is consistent with $\mathrm{ZFC}$ that there is a Mahlo cardinal $\kappa$ so that $\mathbb{M}_{\kappa}$ fails to satisfy the axiom of choice. In fact we may have

Theorems & Definitions (62)

  • Definition 1.1
  • Definition 1.4
  • Remark 1.5
  • Definition 1.12
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Definition 3.2
  • ...and 52 more