The natural consistency-strength conjecture for extensions of ZFC

Let a natural extension of $\mathsf{ZFC}$ mean an extension of $\mathsf{ZFC}$ of the kind considered in the large-cardinal hierarchy, and let $A$ be a large-cardinal axiom (LCA). Its consistency strength is compared by equiconsistency.

Natural consistency-strength conjecture. Any natural extension of $\mathsf{ZFC}$ is either equiconsistent with $\mathsf{ZFC}$ or equiconsistent with $\mathsf{ZFC}+A$, where $A$ is an LCA. Moreover, the consistency strengths of natural extensions of $\mathsf{ZFC}$ are well-ordered.

The conjecture is intended to support Steel’s representation of natural theories by worlds in the multiverse. The qualification “natural” is essential, since contrived consistent sentences can have consistency strengths not captured by the large-cardinal scale.