## Acceptability and Soundness of J-structures.

I would like an example of a J-structure $(J^A,B)$ which is not acceptable and one that is not 1-sound.

Edit:Let us recall that a structure $J^A_\alpha$ is acceptable if for every limit ordinal $\xi<\alpha$. $J^A_{\xi+\omega}\models \vert \xi\vert\leq \vert \tau\vert$, whenever $\tau<\xi$ and satisfies ${\mathcal P}(\tau)\cap J^A_{\xi+\omega}\not \subset J^A_\xi$

A structure $J^A_\alpha$ is 1-sound if the 1-standard parameter is a very good parameter.
(I'm using the notation in Zeman's article in the handbook of set theory)

It is known that $J_\alpha$ are acceptable and sound for every ordinal $\alpha$. Moreover, being acceptable and sound is needed for almost all basic results concerning the $J^A_\alpha$ hierarchy.

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 Could you please give some more background (why you want this, what you have already tried, etc)? – Yemon Choi Dec 7 2011 at 7:47 I wanted to add some context in my partial answer but I'm way to busy this week... Azarel, it would be great if you could add a little context here. I'm not sure all set theorists around here remember the precise definition of acceptable and 1-sound. There is probably some wiki page on the Jensen hierarchy that might explain some of the relevant fine structure. See this meta discussion - meta.mathoverflow.net/discussion/1233/… – François G. Dorais♦ Dec 7 2011 at 14:15

For an amenable $(J,B)$ which is not 1-sound, take a non-constructible real $x$ such that $\aleph_1^L = \aleph_1^{L[x]}$ (and let's say $V = L[x]$ so this is the true $\aleph_1$). Set $B = \lbrace\omega_1+n:n \in x\rbrace$. Then $(J_{\omega_1+1},B)$ is amenable and $x$ is $\Sigma_1(J_{\omega_1+1},B)$ (with parameter $\omega_1$). The $\Sigma_1$-projectum is therefore $1$, so $(J_{\omega_1+1},B)$ cannot be $1$-sound because $J_{\omega_1+1}$ is uncountable and the available parameter set is countable.
 Thank you for your answer. I'm trying to get some intuition into this and it is good to have some examples to get an idea of what can go wrong without these hypothesis. – azarel Dec 7 2011 at 18:21 Note that Schindler and Zeman index the J-hierarchy differently than I do. In their notation, $J_{\omega_1+1}$ should be $J_{\omega_1+\omega}$ and the $\Sigma_1$-projectum is not $1$ but $\omega$. – François G. Dorais♦ Dec 7 2011 at 18:41