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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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  • $\begingroup$ Could you please give some more background (why you want this, what you have already tried, etc)? $\endgroup$
    – Yemon Choi
    Dec 7, 2011 at 7:47
  • $\begingroup$ 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/… $\endgroup$
    – François G. Dorais
    Dec 7, 2011 at 14:15
  • $\begingroup$ 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/… $\endgroup$
    – Brian Rushton
    Feb 2, 2014 at 20:50

1 Answer 1

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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.)

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  • $\begingroup$ 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. $\endgroup$
    – azarel
    Dec 7, 2011 at 18:21
  • $\begingroup$ 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 $\omega$. $\endgroup$
    – François G. Dorais
    Dec 7, 2011 at 18:41

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