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Suppose that $\phi$ is a formula in the language of set theory such that

  1. there are some $n_{1},...,n_{k}$ such that if $V\models\phi(x)$, then $x=(X,R_{1},...,R_{k})$ and $\mathrm{Eq}:X^{2}\rightarrow B,R_{1}:X^{n_{1}}\rightarrow B,...,R_{k}:X^{n_{k}}\rightarrow B$ are mappings such that $(X,\textrm{Eq},R_{1},...,R_{k})$ is a $B$-valued structure such that $B$ is a complete Boolean algebra and if $p$ is a partition of $B$ and $x_{a}\in X$ for $a\in p$, then there is some $x\in X$ with $\|x=x_{a}\|\geq a$ for $a\in p$ ($x$ has the mixing property).

  2. if $V\models\phi(x)$, $x$ is a $B$-valued structure and $y$ is a $C$-valued structure and the structures $x$ and $y$ are isomorphic, then $V\models\phi(y)$ as well.

(Here we say that separated Boolean valued structures $x$ and $y$ are isomorphic if there are bijections $f:x\rightarrow y,L:B\rightarrow C$ so that for each formula $\phi$, we have $L(\|\phi(a_{1},...,a_{n})\|_{B})=\|\phi(f(a_{1}),...,f(a_{n})\|_{C}$.)

Then we say that $\phi$ is a Boolean valued structure formula.

Suppose that $$V^{B}\models\dot{\mathcal{X}}=((\dot{X},\dot{R}_{1},...\dot{R}_{k})\text{ is a relational structure of type $n_{1},...,n_{k}$}.$$ Let $\Gamma^{*}(\dot{X})=\{x\in V^{B}:\|x\in\dot{X}\|_{B}=1\}$ and let $\simeq$ be the equivalence relation on $V^{B}$ where $x\simeq y$ iff $\|x=y\|=1$ and let $\Gamma(\dot{X})=\Gamma^{*}(\dot{X})/\simeq$ (technically $\Gamma(\dot{X})$ is a collection of proper classes, but one may use Scott's trick to represent $\Gamma(\dot{X})$ as a set). Let $\Delta(\dot{\mathcal{X}})$ be the Boolean-valued structure in $V$ with underlying set $\Gamma(\dot{X})$ defined in an obvious way.

We say that a sequence of formulas $\theta;\theta_{1},...,\theta_{n}$ is a determining sequence for a Boolean valued structure formula $\phi$ if whenever $$V^{B}\models\dot{\mathcal{X}}=(\dot{X},\dot{R}_{1},...,\dot{R}_{k})\text{ is a relational structure of type $n_{1},...,n_{k}$},$$ then $V\models\phi(\Delta(\dot{\mathcal{X}}))$ if and only if $V\models\theta(B,\|\theta_{1}(\dot{\mathcal{X})}\|_{B},...,\|\theta_{n}(\dot{\mathcal{X}})\|_{B})$.

When does a Boolean-valued structure formula have a determining sequence? When a Boolean-valued structure has a determining sequence, how does one compute this determining sequence?

I have recently been interested in the relation between structures in $V^{B}$ and how those structures relate to structures in $V$ and I asked a similar question here.

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  • $\begingroup$ You seem to distinguish between $\mathcal X$ and $\dot{\mathcal X}$; could you clarify? $\endgroup$ Apr 28, 2015 at 21:10
  • $\begingroup$ Also, you might clarify what you mean by "isomorphic" for $B$-valued structures, since some authors have a relaxed concept for this (for example, by allowing one to split apart a given name and map it partly to one thing and partly to another, in the mixing style). $\endgroup$ Apr 28, 2015 at 21:23
  • $\begingroup$ In general, $\Gamma^*(\dot X)$ is a proper class (unless $\dot X$ is naming the empty set), and each $\simeq$ equivalence class is typically also a proper class. But one can get a set structure by cutting it down to a "full" set, which is a set of names such that any name that is in $\dot X$ with value $1$ is also equal to an element of the full set with value $1$. By mixing, one can find a full set of names. $\endgroup$ Apr 29, 2015 at 1:40
  • $\begingroup$ Joel David Hamkins. I forgot to put the dots on $\mathcal{X}$. When I wrote $\mathcal{X}$ I meant $\dot{\mathcal{X}}$. I hope I clarified everything you mentioned in the edit that I just made. $\endgroup$ May 3, 2015 at 1:06
  • $\begingroup$ I guess some parentheses are missing in (now) displayed formulas, but didn't try to fix it because I'm not sure of the meaning. $\endgroup$
    – YCor
    Feb 17, 2020 at 13:47

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