5
$\begingroup$

It is well known that Boolean valued models play significant roles for set-theoretic purposes. But how well-studied are Boolean valued models in a more general setting, as models for random first-order languages?

For example, towards the end of chapter 0 of Bell's "Set Theory: Boolean Valued Models and Independence Proofs", Bell gives the definition of a Boolean-algebra-valued $\mathscr{L}$ structure where $\mathscr{L}$ is a first order language whose sole extralogical symbol is a binary predicate $P$:

A Boolean-algebra valued $\mathscr{L}$ structure is a quadruple $\mathbf{M} = (M, eq, Q, B)$, where $M$ is a non-empty class, $B$ is a complete Boolean algebra and $eq$ and $Q$ are maps from $M \times M \rightarrow H$ satisfying, for all $m, n, m', n'\in M$,

$$ eq(m, m)=1, eq(m, n)=eq(n,m), eq(m,n) \land eq (n, n') \le eq(m, n'), Q(m,n) \land eq(m, m') \le Q (m', n), Q(m,n) \land eq(n, n') \le Q(m, n') $$

For any formula $\phi$ of $\mathscr{L}$ and finite finite sequence $\mathbf{x} = <x_1, ..., x_n>$ of variables of $\mathscr{L}$ containing all the free variables of $\phi$, define for any Boolean-valued $\mathscr{L}$ structure $\mathbf{M}$ a map

$$ ‖\phi‖^{M_x}: M^n \rightarrow B $$

recursively as follows:

$‖x_p = x_q‖^{M_x} = <m_1, ..., m_n> \mapsto eq(m_p, m_q),$

$‖Px_px_q‖^{M_x} = <m_1, ..., m_n> \mapsto Q(m_p, m_q),$

$‖\phi \land \psi‖^{M_x} = ‖\phi‖^{M_x} \land ‖\psi‖^{M_x}$, and similarly for other connectives,

$‖\exists y \phi‖^{M_x} = <m_1, ..., m_n> \mapsto \bigvee_{m \in M} ‖\phi(y/u)‖^{M_{ux}}(m, m_1, ..., m_n)$,

$‖\forall y \phi‖^{M_x} = <m_1, ..., m_n> \mapsto \bigwedge_{m \in M} ‖\phi(y/u)‖^{M_{ux}}(m, m_1, ..., m_n)$

That definition, it seems to me, can be easily generalized to a random first order language with other predicate, constant, or function symbols. And Bells mentions that it can be shown that a formula $\phi$ is $\mathbf{M}$-valid for all $\mathbf{M}$ (a forluma $\phi$ is $\mathbf{M}$-valid just in case $‖\phi‖^{M_x}$ is identically 1) iff $\phi$ is provable in classical first-order logic.

So I'm wondering if there are other interesting results on Boolean-valued models as models for arbitrary first order languages. Or in general, how well-studied is the theory of Boolean valued models, as models for random first order languages? How much of traditional model theory (the theory of 2-valued models of first order languages) can be generalized to Boolean valued models? Are there any books or articles on this topic?

Thanks!

$\endgroup$
10
  • 2
    $\begingroup$ Note: this question is cross-posted on MSE. $\endgroup$ – Hanul Jeon Mar 9 at 16:29
  • 3
    $\begingroup$ If I remember correctly, Rasiowa and Sikorski, in their book "Mathematics of Metamathematics" use Boolean-valued models to prove the completeness theorem for first-order logic (in the case of a countable vocabulary). The idea is that a theory has a canonical model with values in its Lindenbaum algebra, and then a quotient by a suitably generic ultrafilter produces a 2-valued model. $\endgroup$ – Andreas Blass Mar 9 at 18:21
  • $\begingroup$ @HanulJeon Yes. Sorry about that! I initially meant to post on overflow but accidentally posted on MSE instead. Shall I delete the one on MSE? $\endgroup$ – Severine Climacus Mar 9 at 20:02
  • $\begingroup$ Somebody may correct me, but I think that general Boolean-valued models are somehow closely related to to sheaf models on Stone spaces. I also think that probability algebra–valued models are also closely related to what are called 'randomizations' in continuous logic. $\endgroup$ – James Hanson Mar 9 at 20:09
  • 1
    $\begingroup$ Continuous logic (in the sense of Chang and Keisler) where the predicates take values in Stone spaces should be equivalent to ordinary discrete first-order logic (possibly without equality). You can always decompose a Stone space–valued predicate into a family of $\{0,1\}$-valued predicates. $\endgroup$ – James Hanson Mar 9 at 20:24
6
$\begingroup$

In 1970th there was more interest on this topic. See for example;

Some aspects of Boolean-valued model theory

Filter Constructions in Boolean Valued Model Theory

On Chang's omitting types theorem in Boolean valued model theory

Eastern model theory for Boolean-valued theories

Also as you mentioned there are connections with sheaves, see

Sheaves and Boolean Valued Model Theory.

$\endgroup$
3
  • 1
    $\begingroup$ Thanks so much! Any idea why the interest on this topic gradually dies out? $\endgroup$ – Severine Climacus Mar 10 at 21:05
  • $\begingroup$ You are welcome. I don't know about it, but this usually happens to many other topics as well, for example, ``model theoretic forcing''. $\endgroup$ – Mohammad Golshani Mar 12 at 8:42
  • $\begingroup$ See also Boolean Types in Dependent Theories. $\endgroup$ – Mohammad Golshani Mar 12 at 8:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.