# Guessing models imply the singular cardinal hypothesis

## TL;DR

This paper establishes new connections between guessing models, the Singular Cardinal Hypothesis, and forcing properties, solving open problems and advancing understanding of set-theoretic principles at uncountable cardinals.

## Contribution

It proves that guessing models are internally unbounded, shows that ISP implies SCH above certain cardinals, and relates forcing properties to covering properties, addressing open questions in set theory.

## Key findings

- Guessing models are internally unbounded.
- ISP(erence) implies SCH above erent cardinals.
- Forcing posets with the -approximation property also have the countable covering property.

## Abstract

In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $\kappa \ge \omega_2$, $\textsf{ISP}(\kappa)$ implies that $\textsf{SCH}$ holds above $\kappa$, and (3) forcing posets which have the $\omega_1$-approximation property also have the countable covering property. These results solve open problems of Viale and Hachtman-Sinapova.