??so suppose that we've got an open subset of the 1-point compactification of N ... ??then let's try thinking of it as "the set of sections of it (thought of as a subterminal sheaf) over _basic_ opens, equipped with the b-valued unary "existence" predicate given by "the basic open over which the section lives", and the b-valued binary "equality" predicate given by "the basic open over which the two sections agree" (??which i guess is just the intersection of the two basic opens ... ???or something ???....) ..." ??? ...... ???or something????....

??so... ??thinking of the basic opens as elements of a boolean ring, intersection corresponds to multiplication ???

???so the collection of basic opens contained within a given open u is ... ??a subset of the boolean ring ... ???the inclusion map being essentially the unary existence predicate??.... and ... ...

??well, there should be a number of different ways of thinking about it, and maybe good to try to keep them somewhat independent so as to be somewhat able to chekc then against each other ... ??or something ... ??

??so.. ??one approach is ...???something like ... ???"coherent geometric theory of a truth value" ... ??or something???...

??first, "coherent geometric theory of a set" ... ???or something ???....

set s equipped with unary and binary b-valued functions ("predicates") ... ??...

"reflexive"

?? "(x=x) <=> (x exists)" ???? ??so existence predicate seems redundant ???


"symmetric"

?? "(x=y) <=> (y=x)" ... ???


"transitive"

?? "(x=y) & (y=z) => (x=z)" ...???...


???then something about ???? "for x distinct from y, (x=y) is strictly falser than (x=x)" ??? or something??? ...??something about "efficiency" or something??? ... ??"irredundancy" ??....

???then something about "subterminal" ... ???... ??"(x=y) <=> ((x=x) & (y=y))" ???or something ????....

??hmmm, so for example, suppose that we have 3-element set {a,b,c} ... ??then suppose we have 9 truth-values aa,ba,ca,ab,bb,cb,ac,bc,cc ... ??? hmmm...

let's start even simpler ... 0-element set ...

???hmm, maybe we also need a sort of "inefficiency" condition ???? .... ???or something ????.... ??somethng about canonical presentation as not very efficient .... ????or something ????......

??what _about_ something about "basis-dependence" (or something ...???...) here ???....

???maybe try ignoring all in-/efficiency constraints for now ... ???

??so 0 generators always gives the initial sheaf ... ???or something ???....

so consider 1 generator a .... 1 truth-value "aa" ... could be anything...

??free boolean algebra on 1 generator ...

??now 2 generators a,b ... 4 truth-values aa,ba,ab,bb ... ??but really just 3 because ab<=>ba ... ??and then transitiveness says... ??what ??... aa*ab=>ab, and ab*ba=>aa, and ab*bb=>ab, and ba*aa=>ba, and ba*ab=>bb, and bb*ba=>ba ... ??? but by symmetricness this reduces to ... ???what ?? ... aa*ab=>ab, and ab*ab=>aa, and ab*bb=>ab, and ab*ab=>bb ...??or something ???...

??xy*yz => xz ...???

??but "p => q" means ...??what ?? ... p*q = p ??? or something ???

so something about "xy*yz*zx = xy*yz" ???

aa*ab*ab=aa*ab ... ???automatic??

ab*ab*aa=ab*ab ??ab*aa=ab

ab*bb=>ab ......................... ????...


??wait a minute, i think that i forgot the "subterminal" condition ... ???.... ??hmmm, which should hopefully bring it down to just freely choosing the existence predicate values, right???

aa bb ... ab <=> aa*bb ...

???so what _about_ simply an ideal ??? or something ???.... ??or complement thereof, or something ... ????....

???some confusion here ...

??something about .... ????collection of truth values that ... ???hmm, well maybe it is just like an ideal ... or something .... ??? ??"put in one global section for each element x in the ideal (??or its complement or something ???), but have its formal existence predicate value be equal to x" ... ???or something ???....

????something about ..... ????homeomorphism type of the open subspace .... ?????or something ??? .... ???hmm, or maybe of its closure or something ?????.....


????something about "system of boolean equations (or something) for which a solution (or something...) sort of amounts to having given stone space as closure (??or something??) of open subspace" ... ???or something ???....

???something about non-/degeneracy and in-/efficiency here ??? or something ...

???hmm, so what about something about case of "atomic truth-value" ?? ... or something ... ???.... ??is there a "the theory (??in what doctrine???) of an atomic truth-value" ???? ... ???or something ???.....