# Tall cardinals in extender models

## TL;DR

This paper characterizes tall cardinals within extender models under the assumption of no inner models with Woodin cardinals, linking tallness to strong and measurable limit of strong cardinals.

## Contribution

It provides a new characterization of tall cardinals in extender models assuming no inner models with Woodin cardinals, connecting tallness to known large cardinal properties.

## Key findings

- Tall cardinals are characterized as either strong or measurable limits of strong cardinals in certain extender models.
- The characterization holds under the assumption of no inner model with a Woodin cardinal.
- The work advances understanding of the structure of large cardinals in extender models.

## Abstract

Assuming that there is no inner model with a Woodin cardinal, we obtain a characterization of $\lambda$-tall cardinals in extender models that are iterable. In particular we prove that in such extender models, a cardinal $\kappa$ is a tall cardinal if and only if it is either a strong cardinal or a measurable limit of strong cardinals.