An Easton like theorem in the presence of Shelah Cardinals

TL;DR

This paper demonstrates that Shelah cardinals can be preserved during GCH forcing and constructs models where GCH-like functions hold while maintaining Shelah cardinals, partially answering open questions.

Contribution

It establishes the preservation of Shelah cardinals under GCH forcing and constructs models satisfying specific Easton functions while preserving these large cardinals.

Findings

Shelah cardinals are preserved under GCH forcing.

Existence of models with prescribed GCH-like functions preserving Shelah cardinals.

Indestructibility results for Shelah cardinals.

Abstract

We show that Shelah cardinals are preserved under the canonical $GCH$ forcing notion. We also show that if $GCH$ holds and $F:REG\rightarrow CARD$ is an Easton function which satisfies some weak properties, then there exists a cofinality preserving generic extension of the universe which preserves Shelah cardinals and satisfies $\forall \kappa\in REG,~ 2^{\kappa}=F(\kappa)$. This gives a partial answer to a question asked by Cody [1] and independently by Honzik [5]. We also prove an indestructibility result for Shelah cardinals.