Meaning of $0^{¶}$ in set theory

Question

I am familiar with $0^{\#}$ and $0^{\dagger}$, but what exactly is the meaning of $0^{¶}$?

Answer 1

$0^¶$ is the sharp for an inner model with a strong cardinal in the same sense that $0^\dagger$ is the sharp for an inner model with a measurable cardinal.

In terms of mice, this is the first mouse containing two overlapping extenders. The effect of this is that, by iterating its top measure throughout the ordinals, you extend the bottom extender in a variety of ways that end up witnessing that the resulting structure is an inner model with a strong cardinal (the critical point of the original bottom overlapping extender being the strong cardinal of the resulting class-sized model).

The critical points you leave along the way (of the images of the top extender, which changes at each step of the iteration) form a family of indiscernibles for the structure, and $0^¶$ could also be described in terms of them, just as $0^\sharp$ can be described by the existence of indiscernibles for $L$.

Martin Zeman's book on "Inner models and large cardinals" is a decent reference for the basic theory of $0^¶$ (and, in particular, for the precise definition of mouse, and for a specific version of fine structure with which to describe the relevant models, and the sense in which a mouse can be least with some property). There is also a brief write up by Koepke, and (harder to locate) notes by Jensen developing the corresponding core model.

MR1876087 (2003a:03004). Zeman, Martin. Inner models and large cardinals. De Gruyter Series in Logic and its Applications, 5. Walter de Gruyter & Co., Berlin, 2002. xii+369 pp. ISBN: 3-11-016368-3.

MR1015308 (90j:03094). Koepke, Peter. An introduction to extenders and core models for extender sequences. Logic Colloquium '87 (Granada, 1987), 137–182, Stud. Logic Found. Math., 129, North-Holland, Amsterdam, 1989.