# 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 '11 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 - tea.mathoverflow.net/discussion/1233/… – François G. Dorais Dec 7 '11 at 14:15
You may be interested in contributing to a proposed Spanish language version of math stackexchange; it could use some input from fluent professors and students: area51.stackexchange.com/proposals/64529/… – Brian Rushton Feb 2 '14 at 20:50

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.
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 '11 at 18:41