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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.
38
votes
Accepted
Could groups be used instead of sets as a foundation of mathematics?
The answer is yes, in fact one has a lot better than bi-interpretability, as shown by the corollary at the end. It follows by mixing the comments by Martin Brandenburg and mine (and a few additional d …
- 42.4k
18
votes
Is Bauer–Hanson’s result “there is a topos where the Dedekind reals are countable” novel?
A first big difference between Brauer & Hansen's result and the one you are talking about is that CZF is a predicative theory (it doesn't have power set/power object) so consistency with CZF doesn't …
- 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
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
13
votes
Accepted
Alternatives to "Sketches of an Elephant" Volume 3
As I said in the comment, this would involve a very large number of different references! (almost one by subsection...)
But to some extent, contributors to the nLab already started doing that and it i …
- 42.4k
13
votes
Accepted
Characterization of Stone-Cech compactifications
I confirme my comment :
$X$ is the stone-cech compactification of a discrete space if and only if $X$ is compact, haussdorf, extremally disconected, and has a dense set of open points.
here is a ske …
- 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
11
votes
Accepted
Free models of finitely presented essentially algebraic theories in elementary toposes?
If you are willing to accept internal argument instead of purely categorical (external) one, a very good reference for this is Palmgren and Vickers' paper: " Partial Horn Logic and cartesian categorie …
- 42.4k
11
votes
Is material set theory conservative over structural set theory?
This will obviously be highly dependent on the concrete theory you are considering. But overall the answer is yes.
The most general version of these results I'm aware of are in Mike Shulman's Comparin …
- 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
11
votes
Accepted
Why is it not possible to define the necessity operator internally $\Box: \Omega \to \Omega$...
As said in the comment, I'm not sure what to add to the paragraph, the point is that in a topos there is no functions $\Omega \to \Omega$ that has the property expected of a neccessity operator except …
- 42.4k
11
votes
Topos properties from coverage conditions
This is a very broad question, we have a huge numbers of such characterization.
But part C of "Sketches of an elephant" contains most of those I know (especially C3). Basically all the notion introdu …
- 42.4k
10
votes
Accepted
Is there an equivalent of the incompleteness theorems/halting problem in category theory?
There is a category theoretic version of the incompleteness theorem originally due to André Joyal that has been unavailable for a long time, but has been written up not so long ago by Joost van Dijk a …
- 42.4k
9
votes
What kind of category is generated by Cubical type theory?
I would say that the question is not even well defined.
Saying that Martin löf type theory with extensional identity types is the internal language of cartesian closed categories with natural number …
- 42.4k
9
votes
Accepted
Do all toposes satisfy the internal Zorn's lemma?
Assuming the law of excluded middle internally your formulation of Zorn's lemma is equivalent to the axiom of choice by the usual argument. Now there are Boolean Grothendieck topos in which the axiom …
- 42.4k