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Results tagged with lo.logic
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user 22131
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
5
votes
Accepted
Which Heyting algbras arise out of some elementary topos which satisfies the ultrafilter pri...
I think your formulation of the ultrafilter principle implies the de Morgan law.
Let $U$ be any proposition, consider the boolean algebra $A$ freely generated by an element $v$.
so $A = \{ 0,1,v,\ne …
- 42.4k
3
votes
Accepted
Simplification in Semi-continuous real ?
Ok I think I finally found an internaly valid proof by my self, so I explain it briefly here in case someone is interested some day :
If $U \in \Omega$ is a subterminal object, you can define the ele …
- 42.4k
3
votes
Accepted
Classification of commutative ring ideal closure operators?
Note that for any such "closure operations" one has $cl(A) = cl(\langle A\rangle)$ where $\langle A\rangle$ denotes the ideal generated by $A$. Hence it can be defined as an operation on ideals.
Now …
- 42.4k
4
votes
1
answer
142
views
Decomposition of an ultrafilter on the fibers of a map
Short version: If I have a map $f:Y \to I$, and $\mu$ an ultrafilter on $Y$, under what condition can $\mu$ be written as a limit/sum/integral of ultrafilters on the fibers of $f$ along the ultrafilte …
- 42.4k
4
votes
1
answer
324
views
Simplification in Semi-continuous real ?
Hi !
I'm considering in a general topos $T$ the object $R$ of lower semi-continuous real (one sided lower non-empty Dedekind cuts, as for exemple in http://ncatlab.org/nlab/show/one-sided+real+number …
- 42.4k
6
votes
Assuming decidable equality but not LEM in HoTT
Decidability of every set implies the law of excluded middle as soon as there is a "subobject classifier".
Indeed, for every proposition $U$, the fact that "$U = \mathsf{True}$ or $U \neq \mathsf{True …
- 42.4k
17
votes
Accepted
Is AC equivalent over ZF to 'every fibration can be equipped with a cleavage'?
Yes, in fact Grothendieck fibration between groupoids are enough.
Let $p:Y \to X$ be any surjection.
We construct the following groupoid $G$. Its set of objects is $X \amalg Y$. Its morphisms correspo …
- 42.4k
7
votes
0
answers
286
views
How "small" can an ordinal be made by forcing?
I know that forcing essentially does not change the ordinals, but by small I mean in comparison with other ordinals whose definition might not be stable under forcing, like the smallest uncountable or …
- 42.4k
7
votes
Accepted
Groupoids as models of symmetric simplicial sets
You can definitely characterize groupoids as presheaves on $Fin_+$ preserving some colimtis (i.e. sending some colimits in $Fin_+$ to limits in Set). In fact Groupoids are the presheaf on $Fin_+$ that …
- 42.4k
8
votes
Accepted
Stone duality for the algebra of Boolean functions such that $f(\top,\dots,\top) = \top$, or...
I might be missing something, but I think you are overcomplicating things.
I clain that your topos classifies the theory $T$ of pairs $(B,\phi)$ where $B$ is a boolean algebra and $\phi : B \to \{0,1\ …
- 42.4k
3
votes
Is there a version of the "infinitary" disjunctive normal form theorem for topoi and slice c...
Edit: I just saw that you were willing to take "every epi split" as an assumption. In this case there is a considerably more classical approach:
Lemma A topos in which every epimorphism split is equi …
- 42.4k
3
votes
Accepted
Products of double-negation sublocales (and probability distributions on them)
For your first question, if $X$ and $Y$ are two boolean locale then $X \times Y$ is boolean only if $X$ or $Y$ is discrete. So unless $\neg \neg A$ or $\neg \neg B$ are discrete, $\neg \neg A \times \ …
- 42.4k
11
votes
1
answer
850
views
Barr's theorem and constructivity?
Barr's covering theorem assert that any Gorthendieck topos can be covered by a Grothendieck topos (even a locale) satisfying the axiom of choice (and hence also the law of excluded middle). Its corrol …
- 42.4k
15
votes
1
answer
2k
views
Topos Without point, from the point of view of logic
I am a little troubled by the following "paradox" :
Let $X$ be a non trivial (Grothendieck) topos without Set points.
We want to look at this situation from the point of view of logic: $X$ classifi …
- 42.4k
12
votes
Example of non-"propositional" local operators on a topos?
Here are examples that are really not propositional in the sense that they are not obtained by combining propositional modalities.
Take a "non-commutative torus"
I.e Takes the circle $S^1$ and makes …
- 42.4k