# Generalizing $0^\#$

Background and motivation: The following theorem is due to Silver:

If there exists a Ramsey cardinal then:

1. For every $\aleph_0 < \kappa < \lambda$, $L_{\kappa}$ is an elementary submodel of $L_\lambda$ .

2.There is a unique closed unbounded class $I$ such that for each cardinal $\kappa > \aleph_{0}$: $\kappa$ belongs to $I$, $I$ has $\kappa$ members below $\kappa$, they generate $L_\kappa$ and they're indiscernibles in $L_\kappa$.

Now, by elementarity and reflection combined with (1), for every $\kappa > \aleph_0$, $L_\kappa$ is an elementary submodel of $L$. Define $0^\#$ as the set of formulas $\phi$ such that $L_{\aleph_{\omega}}$ satisfies $\phi(\aleph_1,...,\aleph_n)$, then by indiscernibility (namely by (2)), $0^\#$ is the set of formulas satisfied in $L$ by increasing sequences from $I$.

Problem: I'd like to generalize the definition of $0^\#$ to arbitrary sets of ordinals. So let $A$ be a set of ordinals.

My naive approach: Change the two assumptions above by replacing each appearance of $\aleph_0$ above by $\sup A$ and by considering structures of the form $L_\kappa[A]$ with the additional relation $A$. Repeating the same arguments as above, for $\lambda$ being the first cardinal above $\sup A$, we obtain the set of formulas $\phi$ such that $\phi(\lambda^+, \lambda^{+2},...,\lambda^{+n}$) is satisfied in $L_{\lambda^{+\omega}}[A]$ (the strucure is considered with the additional one-place relation) as a candidate for $A^\#$.

Questions: My questions are about the following paper by Mitchell: http://www.math.ufl.edu/~wjm/papers/beginning.pdf

In the end of page 13, he gives a completely different definition of $A^\#$, most notably he dispenses with most of the assumptions which are analogous to (1)+(2) above.

Question 1: Is my definition of $A^\#$ flawed? why?

Question 2: What is the justification for dropping most of the assumptions above? Can we still represent $A^\#$ as a set of formulas satisfied by $I$-sequences in some $L_\kappa[A]$?

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I'm not sure Mitchell really dispenses with the assumptions analogous to your (1) and (2). The sentence that begins on page 13 and ends on page 14 contains the assumption that $J$ is a closed proper class of indiscernibles for $L[A]$. I suspect that this implies (with some nontrivial work) most if not all of the analogues of (1) and (2). I further suspect that Mitchell has in mind a metatheory that allows him to talk about truth in proper-class models, rather than using circumlocutions about reflection.
To get started on extracting the analogues of (1) and (2) from Michell's hypothesis, I'd notice that the minimum of $J$ must be above the supremum of $A$. Then, in the structure $(L[A],\in)$, form the Skolem hull of the union of $J$ and the supremum of $A$. The transitive collapse of this Skolem hull won't move the elements of $A$, and it follows that the transitive collapse is $L[A]$ itself. Meanwhile, $J$ has collapsed down to a much nicer class $I$ of indiscernibles (satisfying the same formulas as $J$ in $L[A]$). In particular, I expect $I$ to be still a club in Ord (as $J$ was), to contain all cardinals $\kappa>\sup(A)$, to have $\kappa$ elements below such a $\kappa$, and to generally behave the way you described in (1) and (2).