On Foreman's maximality principle

TL;DR

This paper explores the consistency of the Foreman's maximality principle, showing that under certain conditions, specific forcing notions either add reals or collapse cardinals, advancing understanding in set theory.

Contribution

It proves the consistency of the principle's consequences, including that c.c.c. forcing adds reals and certain closed forcings collapse cardinals.

Findings

Every c.c.c. forcing adds a real under certain conditions.

Certain $ abla$-closed forcings of specified size collapse cardinals.

Consistency results related to Foreman's maximality principle.

Abstract

In this paper we consider the Foreman's maximality principle, which says that any non-trivial forcing notion either adds a new real or collapses some cardinals. We prove the consistency of some of its consequences. We prove that it is consistent that every $c.c.c.$ forcing adds a real and that for every uncountable regular cardinal $\kappa$, every $\kappa$-closed forcing of size $2^{<\kappa}$ collapses some cardinals.