Restrictions on Forcings That Change Cofinalities

TL;DR

This paper explores the limitations of 'nice' forcing notions in changing the cofinality of regular cardinals, demonstrating that such forcings inevitably cause structural disruptions and lack analogues to Prikry forcing.

Contribution

It establishes that forcing which changes cofinalities cannot be too 'nice' and must induce structural 'damage', highlighting fundamental restrictions in set-theoretic forcing.

Findings

'Nice' forcings cannot change cofinalities without structural damage.

No 'nice' analogue of Prikry forcing exists for uncountable cofinalities.

Forcing that changes cofinality necessarily affects stationary sets.

Abstract

In this paper we investigate some properties of forcing which can be considered "nice" in the context of singularizing regular cardinals to have an uncountable cofinality. We show that such forcing which changes cofinality of a regular cardinal, cannot be too nice and must cause some "damage" to the structure of cardinals and stationary sets. As a consequence there is no analogue to the Prikry forcing, in terms of "nice" properties, when changing cofinalities to be uncountable.