Small measurable cardinals

TL;DR

The paper improves the definition of $j$-decomposable systems to better lift elementary embeddings in symmetric extensions, enabling new results about the placement of measurable and weakly critical cardinals.

Contribution

It introduces a refined lifting criterion for elementary embeddings, allowing for more flexible preservation of large cardinal properties in set-theoretic extensions.

Findings

First measurable can be the first weakly critical or Mahlo cardinal.

If first inaccessible is measurable, it has Mitchell order at least 2.

Improved lifting criterion simplifies the analysis of large cardinal embeddings.

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 $2$.