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Small measurable cardinals

Yair Hayut, Asaf Karagila

TL;DR

This work refines the lifting of elementary embeddings to symmetric extensions by weakening the j-decomposability criterion, enabling robust preservation of weakly critical cardinals and extending lifting to weakly compact embeddings. The authors apply this to construct models where the least measurable cardinal is simultaneously the first weakly critical or the first Mahlo cardinal, each relative to a single measurable, and they show that if the first inaccessible is the first measurable, the inner model necessarily has Mitchell order at least $2$. These results illuminate both the flexibility and the limits of choiceless forcing frameworks for arranging large-cardinal hierarchies, and they connect to classical phenomena such as Jech’s results on measurability of $ abla$-levels. Overall, the paper advances our understanding of how lifting criteria interact with symmetric extensions to control the order and interaction of large-cardinal properties.

Abstract

We continue the work from [8] and make a small -- but significant -- improvement to the definition of $j$-decomposable system. This provides us with a better lifting of elementary embeddings to symmetric extensions. In particular, this allows us to more easily lift weakly compact embeddings and thus preserve the notion of weakly critical cardinals. We use this improved lifting criterion to show that the first measurable cardinal can be the first weakly critical cardinal or the first Mahlo cardinal, both relative to the existence of a single measurable cardinal. However, if the first inaccessible cardinal is the first measurable cardinal, then in a suitable inner model it has Mitchell order of at least $2$.

Small measurable cardinals

TL;DR

This work refines the lifting of elementary embeddings to symmetric extensions by weakening the j-decomposability criterion, enabling robust preservation of weakly critical cardinals and extending lifting to weakly compact embeddings. The authors apply this to construct models where the least measurable cardinal is simultaneously the first weakly critical or the first Mahlo cardinal, each relative to a single measurable, and they show that if the first inaccessible is the first measurable, the inner model necessarily has Mitchell order at least . These results illuminate both the flexibility and the limits of choiceless forcing frameworks for arranging large-cardinal hierarchies, and they connect to classical phenomena such as Jech’s results on measurability of -levels. Overall, the paper advances our understanding of how lifting criteria interact with symmetric extensions to control the order and interaction of large-cardinal properties.

Abstract

We continue the work from [8] and make a small -- but significant -- improvement to the definition of -decomposable system. This provides us with a better lifting of elementary embeddings to symmetric extensions. In particular, this allows us to more easily lift weakly compact embeddings and thus preserve the notion of weakly critical cardinals. We use this improved lifting criterion to show that the first measurable cardinal can be the first weakly critical cardinal or the first Mahlo cardinal, both relative to the existence of a single measurable cardinal. However, if the first inaccessible cardinal is the first measurable cardinal, then in a suitable inner model it has Mitchell order of at least .
Paper Structure (8 sections, 17 theorems, 3 equations)

This paper contains 8 sections, 17 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Technical preliminaries
  3. Symmetric extensions
  4. Large cardinal axioms and preservation thereof
  5. Small measurable cardinals
  6. The first weakly critical cardinal
  7. The first Mahlo cardinal
  8. The first inaccessible?

Key Result

Lemma 1

Suppose that $p\in\mathbb{P}$, $\dot x$ is a $\mathbb{P}$-name, and $\pi\in\mathop{\mathrm{Aut}}\nolimits(\mathbb{P})$. Then $p\mathrel{\Vdash}\varphi(\dot x)\iff\pi p\mathrel{\Vdash}\varphi(\pi\dot x).$

Theorems & Definitions (31)

  • Definition 1
  • Lemma 1: The Symmetry Lemma
  • Theorem 2.1
  • Definition 2
  • Theorem 2.2: Theorem 8.4 in Karagila2016
  • Definition 3
  • Theorem 2.3
  • Definition 4
  • Proposition 1: Proposition 3.2 in HayutKaragila:Critical
  • Corollary 1
  • ...and 21 more