Background and motivation: The following theorem is due to Silver:
If there exists a Ramsey cardinal then:
- 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 $L_\kappa[A]$?