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. 



Acceptability and Soundness of J-structures.
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.


(My notes say that this example is from Lee Stanley, but they don't say where I found it. If anybody knows where this is from, please leave a comment.)


