# How far is almost strong compactness from strong compactness

## TL;DR

This paper investigates the relationship between almost strong compactness and strong compactness, providing a negative answer to whether the least almost strongly compact cardinal is strongly compact in general, and clarifies conditions under which they coincide.

## Contribution

It demonstrates that, in general, the least almost strongly compact cardinal need not be strongly compact, resolving a question posed by Bagaria and Magidor.

## Key findings

- Negative answer for the general case
- Conditions under which almost strong compactness implies strong compactness
- Resolution of a question by Bagaria and Magidor

## Abstract

Bagaria and Magidor introduced the notion of almost strong compactness, which is very close to the notion of strong compactness. Boney and Brooke-Taylor asked whether the least almost strongly compact cardinal is strongly compact. Goldberg gives a positive answer in the case $\mathrm{SCH}$ holds from below and the least almost strongly compact cardinal has uncountable cofinality. In this paper, we give a negative answer for the general case. Our result also gives an affirmative answer to a question of Bagaria and Magidor.