??"boolean-valued model" ... ???....

something about ... ??"t-model in not-necessarily-coherent boolean topos" ... ??vs ... ???something about boolean algebra ...

??"boolean space" (or something... "stone space" ...) vs "boolean locale" ... ???....

??something about which of these is relevant to "forcing" ... ??and so forth ...

???something about nonstandard analysis ... "ultrapower" as involving adjoining of generic element, sort of...

???bit about ... "ultrapower as still elementarily equivalent to original" ... ??though something about slice category... extra constant ... ??? ... anyway, vs something about "boolean-valued model" and "forcing" ... or something ...

??something about ... with "boolean-valued model", is "the boolean algebra" in question just an external (or something...) such, or is it an object in a topos (??such as the underlying topos of a sufficiently good epsilon-universe ?? ... ??or something???...) ... ???

??remembering some confusion (and so forth...) about something like this ...

??something about ... ??the truth-value object in a boolean topos ... such as a boolean localic topos ... ??

??what about something about ... ??instead of passing to double-negation topos, considering internal double-negation boolean algebra of truth-value heyting algebra ??? ... or something ... ??... and so forth ... ??...

??hmmm... wpa on "boolean-valued model" specifically says something about _complete_ boolean algebra .... ??? .... hmmm....

??so what _about_ non-standard analysis (or something... and so forth ...) as involving a different sort of "boolean-valued model", based on just plain boolean algebra ??? ... ???or something ??? .... ??or is that really a good way of thinking about it ... ??? ... not sure ... ???...

hmm ... ??...something about ... maybe actually ... ??completely distributive boolean algebra... simply power set boolean algebra ... ??but what should we think of it as "being treated as" ?? ... ??boolean algebra... ??complete (or something...) boolean algebra ... ??completely distributive boolean algebra ... ?? ....

??for one thing, is the "boolean-valued model" supposed to preserve infinite disjuctions??? ... or something ... ??wouldn't particularly expect to see such a requirement (??or maybe it's obvious that we won't see such a requirement?? ... ??would be too restrictive ???), but then why bother mentioning completeness??? ... or something ... ??? yes, could have to do with "examples showing up in practice" or something ... but... still confusion ...

????hmmmm, they say this:

"The completeness of the Boolean algebra is required to define truth values for quantified formulas."

...really have to think about that ... not at all sure that i get it yet .... ????????.....

well, i sort of get where they're coming from (was trying not to read their detailed discussion yet but accidentally saw word "supremum" ...), but i still have to think about it a lot ... ??.....

??something about ... ???attitude of something about "interpretation preserves certain relationships", vs "interpretation is given by values at generators, with generating process needed in codomain to flesh it out ..." ...??or something ... need to think about this ... is there really some nice lawveresque way of exploiting adjoint functor interpretation of quantifiers here ... and so forth ... ????....

??something about idea of "coherent boolean topos" approach to classical first-order theories ... ???as making it seem like completeness isn't very relevant here (??...) ... ???or something ???...

??something about ... ??"quantifier as infinitary operation vs a unary" and something about "actual variable vs formal variable" .... ???or something??? ....