Arghhh!!! :grimace: This is an example of intuitionistic dependently typed theory, corresponding to a non-trivial topological space. The previous one ...
WOW!!! I understand only partially what the terms mean: analytical continuation of a complex function has something to do with chaotic systems? Did I ...
To relate to something that probably you know better: the Riemann zeta function is related to the distribution of prime numbers: why complex functions...
Yes, exactly! Look for example at this: https://homotopytypetheory.org/ Well, not only is useful, but if you find a relation between apparently comple...
That's an example of the relation between logic and topology. A fiber bundle (topological space) can be interpreted as a set of propositions speaking ...
Here's the example of a fiber bundle that I promised. The BASE SPACE is constituted of 3 propositions (propositios are types): P1 := {A;B;C} P2 := {C;...
OK, I understand what you want to do. But in the case of fiber bundles you don't define the topology of the total space in terms of the topology of th...
The problem with your interpretation is that you don't consider variables. You build a model of propositional calculus ( https://en.wikipedia.org/wiki...
The underlying set is the set of all propositions. The fibers are sets of elements of our model. OK. Yes, unfortunately I am not able to follow your p...
:smile: Thanks for trying to help! But it's not so simple... I am afraid @"fishfry" has chosen the most complicated way to "build" a topos: the one th...
Part one: C is a category ( like A is an abelian group ) Part two: for each pair of objects of A, B there is a "product object" P ( like for each pair...
Yes, well, the point is that you cannot "count" the objects of a category. You cannot distinguish between isomorphic objects. There is no "equality" r...
That's pretty standard old-fashioned model theory and first order logic (the topology is irrelevant: forget about open sets and take simply the set of...
Yes, but that correspondence is evident only in a dependent type theory, where you can make sense of the topology defined on your set of propositions ...
Here's the explanation in straightforward terms: a topos is an "extension" of the category of sets. ( probably it should have been called "setos" :gri...
About the randomness of sequences, I think a good definition is the following one: a sequence is random if it cannot be generated by a program shorter...
Hmm... :chin: You want a topological space for classical logic. OK, a topological space is a set of all subsets of an "universal" set. - The elements ...
It wasn't inaccurate, it was a particular case, as you usually do when you give an example.. Yes, but you can easily find the precise definitions on W...
A subobject classifier is a pair of an object and an arrow {Omega, "true": T->Omega} with the following property: every monomorphism m: A->B in the ca...
I know Coq. And I know type theory because it's the logic implemented in coq. And type theory is the internal logic of a topos. I read some books abou...
Nat is the type of natural numbers. Types are represented by objects of the category, derivations are represented by the arrows. That's in the book fr...
Well, if you want the exact definition of sheaf I can copy it from the book on category theory that I posted you yesterday. I don't know all possible ...
OK, you are right! What I wanted to point out is that the sentence "there is an uncountable number of sequences", when expressed in the forall-exists ...
Let me just give you just some examples: "x >= 3" is a fibration from the object Nat to the subobject classifier Prop. The proposition "5 >= 3" is a f...
OK, I'll start from the example: 1. Let's call Unit our terminal object. 2. Let's call Prop the object that is part of our subobject classifier (usual...
I noticed that I was mentioned here, so I started to read this thread from the beginning. And I realized that probably this is the perfect example to ...
No, I like "normal" mathematics: no computers involved. But having a theorem-prover as Coq to be able to verify if you can really write a proof of wha...
Yes! I should have added a formal definition, but I have an aversion to writing symbols on this site :confused: I added a link with a clear picture, I...
I know about measure theory since when I was at high school (I always liked that stuff), and I heard that sentence about the probability to find a rat...
A sheaf is a topos at the same way as a set is a topos: it's the "trick" of the Yoneda embedding! :smile: do you understand now? (sorry: bad example.....
Yes, I work as a programmer. And yes, a formal proof completely misses the essence of a proof: it's "meaning". I don't beprove lieve mathematics is ch...
OK, that's the "like I'm five" explanation: A topos is a category that "works" as the category of sets, but is not built using a set of rules that ope...
Re-reading your question I just realized that probably my answer about homotopy type theory could be misleading: this is really a weird and interestin...
Yes, it seems very strange! Here's the "quick and dirty" explanation: A sheaf S over a topological space X is a "fiber bundle", where the fibers over ...
Probably you think that I completely missed the "meaning" of what a mathematical proof is. But that's the way computers (formal logic systems) see pro...
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