# Measurable cardinals and good $\Sigma_1(\kappa)$-wellorderings

## TL;DR

This paper explores how the existence of measurable cardinals affects the possibility of defining wellorderings of power sets of infinite cardinals using simple formulas, revealing deep interactions between large cardinal axioms and definability.

## Contribution

It demonstrates that measurable cardinals prevent certain wellorderings at specific cardinals and identifies measurability as the minimal large cardinal property impacting such definability.

## Key findings

- Measurable cardinals prevent wellorderings at certain cardinals.
- Such wellorderings exist at all other uncountable cardinals in the minimal model with a measurable.
- Measurability is the smallest large cardinal property affecting these wellorderings.

## Abstract

We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $\kappa$ with the property that the collection of all initial segments of the wellordering is definable by a $\Sigma_1$-formula with parameter $\kappa$. A short argument shows that the existence of a measurable cardinal $\delta$ implies that such wellorderings do not exist at $\delta$-inaccessible cardinals of cofinality not equal to $\delta$ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a measurable cardinal. In addition, we show that measurability is the smallest large cardinal property that interferes with the existence of such wellorderings at uncountable cardinals and we generalize the above result to the minimal model containing two measurable cardinals.