The core model for almost linear iterations

TL;DR

This paper explores the existence of a core model under the assumption that a certain large cardinal axiom (0^h) does not exist, establishing the existence of the core model K in ZFC with minimal large cardinal assumptions.

Contribution

It proves the existence of the core model K assuming the non-existence of 0^h, refining the understanding of the minimal assumptions needed for core model existence.

Findings

If 0^h does not exist, then any normal iteration tree is almost linear.

The core model K exists in ZFC + '0^h does not exist'.

This result identifies the weakest anti-large cardinal assumption for core model existence.

Abstract

We introduce 0^h (0^handgrenade) as a sharp for an inner model with a proper class of strong cardinals. If 0^h does not exist then any normal iteration tree is "almost linear." We exploit this fact to prove the existence of the core model K in the theory "ZFC + 0^h does not exist." (As of today, non-0^h is thereby the weakest anti large cardinal assumption under which K can be shown to exist in ZFC. In this sense we improve earlier work of Dodd, Jensen, Mitchell, and - partially - Steel.) We indicate that our paper provides the last step for determining the exact consistency strength of a statement in the Delfino problem list.