A fiber bundle is like the collection of tangent planes to a sphere. Somehow, one can replace the tangent planes with logical structures of some sort, and the points of the sphere with .... something, and intuitionist logic drops out. Perhaps it's not explainable in elementary terms. But I couldn't relate what you wrote with any attempt to clarify this point. — fishfry

The usual intuition is more like an "airbrush" ( https://en.wikipedia.org/wiki/Fiber_bundle ). The fibers are seen as stick wires coming out from a common surface; they are separated from each other.

Not clear to me. I literally and honestly did not understand what you said in this post. Perhaps it's a lost cause. — fishfry

Well, OK, never mind. However, the book that I gave you the link is very clear and contains proofs and exact definitions. Surely that's easier to understand than my explanations...

I'm afraid I share Metaphysician Undercover's misgivings about this remark. I understand the categorical viewpoint of sets, but I would not characterize that viewpoint via this particular way of phrasing it. — fishfry

All right. Fair point.

I found a paper that indicated the the fibers are "L-structures." Not too sure what those are, or what the base set is. I'm not sure I entirely believe it's a discrete topological space. I'm thinking you've probably explained this point to me several times over but I still don't get it. My apologies for giving you a hard time out of frustration at my inability to understand how fiber bundles can be used to model logical structures. — fishfry

I don't know what are "L-structures", but I think I know what's the source of misunderstanding: the words "discrete" and "continuous" used to refer to finite structures. In my example the "space" of the model is made of 12 points, but it's NOT a discrete space: not all subsets of the set {1...12} are open sets BY DEFINITION. The definition of which sets are open is arbitrary: the only required conditions is that it has to include the empty set, the full set, and all possible unions and intersections.

You should see the topology as a kind of "blurring glass" that is put over the set {1...12} and does not allow you to distinguish the individual points: you can see groups of points, but not individual points. Think of the set of real numbers when they are interpreted as results of physical experiments: you explained this to me very clearly: you can never get a real number as the result of an experiment (and you can't split a physical sphere in distinct points as in the Banach-Tarsky theorem).
The same thing can be true for the set of 12 points in my example: you cannot distinguish the point 1 from the point 2, because there are no open set {1} and open set {2} in the topology.
Of course, the most interesting cases of open sets are infinite sets (as real numbers), not finite ones as in my example. But I especially chose a finite set to make it crystal clear: topology is not about the cardinality of the "universe" set.

Yes, but that's not mathematics! The distinction of which concepts are more "fundamental" is very useful to "understand" a theory, but it cannot be expressed as part of the theory. Mathematical theorems don't make a distinction between more important and less important concepts: if a concept is not needed, you shouldn't use it. If it's needed, you can't prove the theorems without it.

P.S. That's a very important point to understand: the words used in mathematical sentences are not chosen at random: they are carefully chosen to give some "intuition" of the things that we are speaking about. However, you cannot use that intuition in proofs. Proofs have to be completely "formal": they have to be valid even if you substitute the words with random strings of characters.

I see mathematical axioms expressed in plain English. — Metaphysician Undercover

There is a way to translate any mathematical proposition (or axiom) into plain English, but there is no way to translate any English proposition into a mathematical proposition: formal languages are more limited than natural languages.

It's not true that words are worth nothing in mathematics, because the axioms are written in words. My demonstration was a proof, a logical proof that a set cannot be more fundamental than its elements, because that creates an infinite regress. If you are satisfied with an infinite regress you have an epistemological problem. Such mathematics is not supported by sound epistemology. — Metaphysician Undercover

I meant words in plain english language (or in another natural language): you have to use a formal language to express mathematical theorems.

What do logic and topology have to say about each other?

Specifically; if a logic has a model is there a correspondence between a topological space on the set which models it and how proof works in the logic? — fdrake

(continuation: correspondence between a topological space and how proof works in the logic)

The rules of logic should be valid for ANY topology and ANY subset of points. So, for example, they should work even for a discrete topology.

Now, for intersection and union everything works fine, since the intersection of two open sets is again an open set, and the union of two open sets is again an open set. But the complement operation does not preserve the openness, so it cannot be a primitive operation in an algebra of open sets.

Well, the "trick" to make it work is very simple: do NOT considered it as a primitive operation but define "complement of A" as "the largest OPEN set X such that the intersection of X and A is empty" (the largest open set with no points in common with A, but we don't need to speak about points)

With this definition the two logics are exactly the same in the limit case when all subsets are open, but the second one (intuitionist logic) is weaker: every intuitionist derivation is even a standard derivation, but there are boolean logic derivations that are not intuitionistic derivations. The difference is due to the fact that intuitionistic logic does not "see" the points that are not part of open sets (the points on the border of an open set, or even isolated points). If you take a look at the example with fiber bundles that I posted a couple of days ago, you can see what I mean: even if you think of your model as a set of points, you can't speak about single points using this language: every eventual single point not included in an open set is simply "ignored".

The effect on the logic rules, as @fishfry pointed out, is that double complement (double negation) does not give you back the original set: if you had a set A that was not open nor closed (let's say an open set plus some part of the border), the first complement operation gives you the complementary OPEN set; and then the second negation gives you back A WITHOUT BORDER (that now has become open. Taking again the complement a third time, now you obtain the same result as taking it once, and so on.

The effect on the rules of logic (one of the effects) is that the excluded middle is not valid rule. But if you add the rule of excluded middle to intuitionistic logic as an additional axiom you don't obtain an unsound system: you simply get back boolean logic. This is equivalent to choosing as topology of the space the discrete topology. It's an additional assumption: intuitionistic logic works for any topology; boolean logic, instead, works only in the particular case of the discrete topology.

Aristotle demonstrated this premise, that the Form, as a universal type, (what you call "the set") is more fundamental than its elements, leads to an infinite regress and is actually impossible, therefore false. — Metaphysician Undercover

What I wrote is only an idea, that (in my opinion) is important to understand the "meaning" of a theory, but from the point of view of mathematics all explanations that you can give by words are worth nothing: at the end, the only thing that counts in a mathematical argument are proofs. If what you say cannot be proved, it's not mathematics. I know, neither of us presented any proof of what we said here, but we are on a philosophy forum here, right? :wink:
What I want to say is that your argument "... leads to an infinite regress and is actually impossible, therefore false" (Aristotle's argument) would not be accepted as a valid prove in today's mathematics.
In mathematics you are free to "invent new worlds" (I believe this is what Grothendieck was saying about his work), but you have to do it using proofs that are rigorous enough to be accepted by peer reviewers.
Then, you can discuss if what you invented is "important", and what's it's "meaning".

Making the "One" the most fundamental resolves the inherent contradiction of having the empty set as fundamental. The empty set is inherently contradictory because it is something, an object, which at the same time must be nothing. — Metaphysician Undercover

The empty set is only an abstract construction defined by a set of axioms that has nothing to do with the "One" of ancient Greek philosophers.

What do logic and topology have to say about each other?

Specifically; if a logic has a model is there a correspondence between a topological space on the set which models it and how proof works in the logic? — fdrake

I see that there is a misunderstanding between us on what it means "a logic has a model".

A logic is a bunch of rules that describe how you can build sentences that speak about "something".
What I call model is that "something". For example, the real numbers can be the model. The model is the thing that we are speaking about. The rules of logic have nothing to do with it! If we speak about the waves of the sea, then "the set of all waves of the sea" is the model. It is "the real thing" that we are speaking about.

Now, the essential change in the point of view that allows you to see the correspondence between topology and logic it this one: consider sets to be more "fundamental" than their elements.
So, if our model are the real numbers, the sets of real numbers are more "fundamental" than the single real numbers. If you think about it, that's what boolean algebra does: boolean algebra speaks about sets and operations between sets (union, intersection, complement): you build sets starting from other sets, without mentioning their elements.

From this point of view, we have an "universe" set (in our example the set of all real numbers), and the set of all sets of real numbers (the powerset of the "universe"), and a boolean algebra defined on the powerset of the "universe".

Now, we can generalize logic by substituting the powerset of the "universe" with a topological space. A topological space is in general defined as the powerset of the "universe", plus a choice of which elements of the powerset are "open" (this choice of the open sets is what's usually called the "topology").
A topological space, then is a generalization of the powerset of the "universe": instead of considering as the fundamental elements of your algebra the full powerset of the "universe", you consider as your fundamental elements the open sets of the universe (in our example, the open sets of real numbers). The algebra built on the open sets is a Heyting algebra. The algebra built on the full power set of the universe is a Boolean algebra. Of course, you can consider the Boolean algebra as a particular case of Heyting algebra by choosing as open sets all the subsets of the "universe" (the full powerset of the universe). This is what is called a "discrete" topology.
So, logic built as an algebra based on a topological space is a generalization of the logic built as an algebra on the powerset of the universe, at the same way as a topological space is a generalization of the powerset of the "universe".

( I'll describe the part related to proofs the next time )

Arghhh!!! :grimace: This is an example of intuitionistic dependently typed theory, corresponding to a non-trivial topological space. The previous one was an example of first order logic with set theory, since you didn't want type theory. And THERE IS NO NON-TRIVIAL TOPOLOGICAL SPACE CORRESPONDING TO FIRST ORDER LOGIC WITH SET THEORY. Than in that example, the topology was irrelevant! @fdrake, did you understand?
If yes, can you please try to explain this in a better way? I don't even have much time for this, sorry. I have even to go to the hospital for a couple of days next week.

I showed that convergence of complex limit periodic continued fractions useful as functional expansions could be accelerated by employing a feature of dynamical systems: attracting fixed points (Proceedings of the AMS). And could be analytically continued by using repelling fixed points (Mathematica Scandinavica and Proc. Royal Norwegian Soc. of Sci. & Letters). There are deeper meanings here by locating these concepts in theory of infinite compositions of complex functions. — jgill

WOW!!! I understand only partially what the terms mean: analytical continuation of a complex function has something to do with chaotic systems? Did I understand correctly? Why didn't I even here anything about this? And YOU proved it? Now I am impressed! Really! ( or maybe I didn't understand a thing of what you just said... )

To relate to something that probably you know better: the Riemann zeta function is related to the distribution of prime numbers: why complex functions should have something in common with integer arithmetic?

Yes, exactly! Look for example at this: https://homotopytypetheory.org/

Well, not only is useful, but if you find a relation between apparently completely different areas of mathematics, maybe those concepts have in some way a deeper meaning.

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 about some model. They are completely different concepts, but the algebraic structure is the same.
(The logic however, is not the standard logic of set theory)
P.S. we lost the topic of the thread a long time ago... :grin:

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;D;E}
P3 := {E;F;A}

Capital letters are the "constructors" of the types, that you can see as the simplest possible kind of "rules" of our logic.
- to prove P1 you can use A, B or C (a proof of P1 would be written as "A: P1")
- to prove P2 you can use C, D or E
- to prove P3 you can use E, F or A (the order is irrelevant)

From the topological point of view, of the space of the propositions is made the following open sets:
P1, P2, P3, the empty set, the set {A;B;C;D;E;F}, and all possible unions and intersections of P1, P2 and P3. (the letters are the points of the base space)


The total space is made of the hours of the day, from 1 to 12 (no distinction between morning and afternoon to make it simpler).
Our model is an object made as a clock, with 12 hours painted in circle, and an arrow indicating a "set of hours". Let's say that the arrow in general does not indicate a precise hour, but a set of them.
The possible sets of hours that can be indicated by the arrow are the following ones:
{1;2}, {3;4}, {5;6}, {7;8}, {9;10}, {11;12},
{1;3}, {2;4}, {3;5}, {4;6}, {5;7}, {6;8}, {7;9}, {8;10}, {9;11}; {10;12}; {11;2}; {12;1} (the last two sets are not a mistake)

We define the topology of the total space in the following way:
- all the sets of the previous list are open sets;
- the empty set and the set of all hours are open sets;
- all possible intersections and unions between sets of the previous list are open sets.
No other subset of hours is an open set.

Now, we define our projection function X: a map from the total space of the hours of the clock to the base space of the propositions describing the result of our experiment:
1 => A; 2 => A; 3 => B; 4 => B; 5 => C; 6 => C; 7 => D; 8 => D; 9 => E; 10 => E; 11 => F; 12 => F.
( notice that this is a continuous function )

The inverse image of X is the following one:
A => {1;2}; B => {3;4}; C => {5;6}; D => {7;8}; E => {9:10}; F => {11; 12}

Let's check the homotopy type of our fibration.
We start from point A - 1 and follow the continuous path A-B-C-D-E-F-A in the base space.
- when you are on B, you can move only to 3 on the total space because there is no open set containing both 1 and 4 (the rule, of course, is that you have to follow a continuous path in the total space)
- then, from B - 3, you are forced to move to C - 5 (because {3,6} is not an open set)
and then, continuing in this way, we see that the only continuous path (the only possible section of the fiber bundle) is the following one:

A-1; B-3; C-5; D-7; E-9; F-11; A-2; B-4; C-6; D-8; E-10; F-12; A-1
We see that the fiber bundle is not trivial (double covering of the base space).

Now, let's come back to the logic interpretation.
The meaning of our propositions is given by the inverse image of X (let's call it Y).
So, P1 means that our arrow points to the set of hours {1;2} or {3;4} or {5;6} (then, it means "we are in the first third of the day")
P2 means "we are in the second third of the day"
P3 means "we are in the third third of the day"

But now, let's check which propositions we can form starting from P1, P2 and P3.

"P1 and P2", for example, means that the arrow is on {5;6}, corresponding to the open set {C} of the base space.
But there is no way to say that the arrow is on {3;4}, because the {B} is not an open set in propositions space. B is a point of the base space, but the propositions correspond to open sets, not to single points. And not all subsets of {A;B;C;D;E;F} are open sets!
You see, the open sets have meaning but not the points of the space
Or, better, the points of the base space correspond to proofs of our propositions, but what meaning can you give to the points of our model? (the hours of the clock). Our clock's arrow is too "fat" to distinguish between single hours, so only sets of them are meaningful hour indications. So, for example, you'll never be able to say the difference between 1 o'clock and 2 o'clock using our logic, even if in the topological space of the clock they are different points, and the space is "made" of points in our case: this is standard point-set topology.


[OK, it's just become way too long...] But at least NOW WE HAVE A CONCRETE EXAMPLE.

P.S. The types the proofs in this example are the simplest possible example of inductive types.
But of course this is not all: for example, it would be impossible to represent the set natural numbers in this way. But I cannot write a book on type theory on this site....

I was hoping you'd be able to tell me, as it seems defining what a topology is in terms of the logic is precisely where the missing intuition is. I assumed there was some topology on the space of propositions, and tried to see if "pushing back" the open sets of the topology on ×KΩ×KΩ through the interpretation II made sense to you.

Something like, if we have a logical connective's truth function ff from ×KQ→Ω×KQ→Ω, and we glue together open sets on ×KΩ×KΩ 's discrete topology through the fibre {I(a)=1=f(c)|a∈×KQ,1∈Ω,c∈×KΩ}{I(a)=1=f(c)|a∈×KQ,1∈Ω,c∈×KΩ}. I guess inducing a topology by pulling back a topology through a continuous function and a connective (dunno if that works at all). — fdrake

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 the base space. You assume a preexisting topological space E (the total space), and a preexisting topological space B (the base space), and a continuous function P from E to B, and then you define the fibers as inverse images of P.

It perhaps doesn't make much sense. Do you know the analogous construction for classical (or intuitionist) propositional logic? — fdrake

Yes, it's a Heyting algebra ( https://en.wikipedia.org/wiki/Heyting_algebra ) is the analogous of boolean algebra for intuitionistic propositional logic.

So we don't have to deal with the interpretation being a complicated set valued function. It should be in there somewhere and easier to talk about to exhibit a connection between a topology on the logic and a toplogy on the Omega product. — fdrake

But that's the whole point! You have to be able to talk about open sets to make sense of a logic that allows the existence of open sets not "built" as sets of points. The logic that we are talking about is (for example) the one that allows the existence of infinitesimal numbers that are not non-zero but whose square is zero.
Well, now that I think about it, I heard that you can see the modal connective of modal logic as an arrow of the form Omega to Omega (from propositions to propositions, the same as negation), and that this arrow is (represents) a topology on Omega (the set of truth values). So, the modal connective could be interpreted as "it is locally true that". But for me this is a little too "abstract" :smile:; and first of all, I don't know modal logic! Maybe there is a way to make a precise sense of your idea, and I just don't know about it.

The problem with your interpretation is that you don't consider variables. You build a model of propositional calculus ( https://en.wikipedia.org/wiki/Propositional_calculus ) by assigning to each elementary proposition a truth value, and not of predicate logic, where you assign to each variable of an open formula (meaning: a formula without quantifiers) a value of the set that you consider as your domain of discourse. What you obtain in this way is a boolean algebra of propositions, where every elementary proposition is a point of your universal set. In this way the pre-image of 1 (or "true") is simply the set of all true propositions. So you get a discrete space of points split in two equivalence classes: true propositions and false propositions. What is missing to have first order logic is the interpretation of variables and quantifiers. In your model the topology of omega is not used, and there is no topology defined on the space of all propositions (how do you define the set of open sets of propositions?), so you cannot define continuity either.
Or maybe did I miss something?

Well, I know what a fiber bundle is so if you claim something is a fiber bundle you could just explain what it is that's the fiber bundle. What is the underlying set, what are the fibers above each point, etc. But maybe there's too much of an explanatory gap and we're at a point of diminishing returns. — fishfry

The underlying set is the set of all propositions. The fibers are sets of elements of our model.

This is not about introduction to first order logic. This is about an explanatory gap. The topology is not irrelevant if you claim to have a sheaf. Perhaps we're done. — fishfry

OK.

We're talking past each other. And this is not about cardinalities at all since neither proper classes nor categories (in general) have cardinalities. But I think between what you know and what you're able to explain, and what I know and what I'm able to understand of what you're saying, we have a gap that's not getting bridged. — fishfry

Yes, unfortunately I am not able to follow your plan.

And why would that be a problem? A group is a category with one element. — fishfry

It is not a problem. It's only an example: in general a category can have any number of objects, but a cartesian-closed category must have an infinite number of objects: the additional condition that all binary products exists implies a restriction on the possible number of objects.

If we extended the topology to the product space ×JΩ×JΩ, I thiiiink this ends up being the discrete topology? But then it's also the set of all truth table rows of propositional logic formulae containing at most |J||J| propositions (well, so long as they don't contain duplicates...). — fdrake

: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 that Gothendieck come up with at a time when (I believe) the notion of category didn't exist yet. And, what's worse, he wants to do it as a model of standard set theory. There is no model of standard set theory that can be seen as a sheaf! And this is for various reasons:
1. A sheaf is a continuous contravariant map from open sets to sets. The open sets should be thought of as the open sets of our model: (open sets of real numbers, for example, if we consider first order logic speaking about real numbers). But first order logic does not speak about sets of elements, but only (at most) about tuples of numbers! (forall x,y there exists z such that ...). To speak about sets you need higher order logic.
2. Even if you consider higher order logic, the algebraic structure corresponding to boolean logic (as you said) is a boolean algebra. Now, the topology that you get from a boolean algebra is a discrete topology: every set is both open and closed. This topology is the only particular case that does not contain any "information" about the "connectivity" of (the topological information about) the space: every set is simply a set of "isolated" points. Maybe I wasn't able to explain this: you can think of a topology as two different pieces of information: 1. The set of all subsets of a given "universe" set; 2. An (arbitrary) choice of which ones of these subsets are considered to be "open". In a discrete topology all subsets are open; so you don't really need a topology in case of a boolean algebra: it's enough to consider the set of all subsets of the universe, and that's what you do in boolean higher order logic.

Instead, if you consider a topos from a categorical point of view (a topos is simply a category with some additional structure), the set-theoretical operations (intersection, union, complement) are a Heyting algebra, quantifiers are an adjunction between categories, and open sets are naturally represented as types in type theory, that can be seen (in my opinion) as a nice generalization of high order boolean set theory, where you can make sense of the information related to the topology of the domain (not all subsets are open). Everything is much simpler to understand than Grothendieck's construction based on sheaves.

I am sorry if I am not able to explain this in a clearer way...

Minimal set of universal properties. That might be over my head. I know what universal properties are in terms of defining things like free groups, tensor products, and the like. I'm ignorant of what it would mean to select for certain universal properties. Or I'm not understanding you. — fishfry

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 of elements (a,b) of A there is a product element: A is a ring ) ( omitting other needed properties, of course... )
Adding properties to the category I add structure! For example, each pair of objects has a product, the set of objects has to be infinite (pairs made of other pairs recursively). In general, without this requirement, a category may even be made of 3 objects and 4 arrows...

To be fair, that's not it. I feel like you didn't address my specific question or expand on your remarks about fiber bundles, fibrations, and the n-tuples of real numbers taken as synonymous with propositions. Those are the kind of "bread and butter" things I'm trying to understand. But like I say I'm perfectly ok with that, because I got my money's worth from discovering Mac Lane. — fishfry

OK, as I said, I'll get to fiber bundles on the next episode.. :smile:

I've read different points of view. For example the question arises, is the category of sets the same as the proper class of all sets? Well, not exactly. I've heard it expressed that the category of sets has "as many sets as you need" for any given application. But it's not exactly synonymous with the class of all sets. That's my understanding, anyway. — fishfry

Yes, well, the point is that you cannot "count" the objects of a category. You cannot distinguish between isomorphic objects. There is no "equality" relation defined on the set of all objects. How can you decide what's the cardinality of the set of all objects if you cannot associate them with another set? (no one-to-one correspondence possible between elements. Only equivalence makes sense, not equality!)

Likewise you claimed that the collection of n-tuples of real numbers with the discrete topology can be associated with propositions. You have my attention with that example, I just need to see the rest of it. — fishfry

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 all subsets of a given set R). I noticed that other people on this site were starting some kind of "introduction to first order logic" thing. Maybe they can help to make clear this part.

You've said proofs and propositions are like fiber bundles or sections. And at one point you used the word "fibration," which is a very specific thing in topology. I was hoping you would either explain those remarks with laserlike clarity, as if you were writing an exam; or else agree that for whatever reason we can't do that. — fishfry

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 (only open sets are propositions). In standard logic you cannot make sense of the topological structure of the space: no distinction between open and closed sets. All sets are both open and closed. That's the reason why taking the complement of the complement is an identity (boolean logic!). How can I show you the correspondence with dependent type theory without explaining dependent type theory?

My latest understanding is that a topos is like a generalize universe of sets. You can have one kind of universe or a different kind of universe depending on the rules you adopt, but all set-theoretic universes fit into the topos concept.

In fact the category of sets IS a topos. That's very helpful. Sometimes the simplest examples are the ones to start with. — fishfry

Yes!

You said a while back, in a remark that started this convo, that you can do logic in a topos. I'm curious to understand that in straightforward terms. It doesn't seem that difficult once you know how the mathematical structures are set up — fishfry

Here's the explanation in straightforward terms: a topos is an "extension" of the category of sets.
( probably it should have been called "setos" :grin: )
First of all, then, you have to know how you can do logic in the category of sets.

The category of sets is the category that has as objects ALL the sets and as arrows ALL the possible functions from any set to any set.

We just know how to do logic in ZFC set theory ( :confused: or should I start from the standard first order logic in ZFC? ), so we could do exactly the same thing here, but there is a problem: there are no "elements" inside an object ( the sets are represented by objects, but objects in category theory are a primitive notion: they are defined axiomatically, and not by describing how to build them starting from elements ).
However, there is a way to "represent" all the the logical operations of set theory ONLY in terms of the objects and arrows of a category. Here's the mapping:
- The empty set is represented by the initial object of the category
- The singleton set (all singleton sets are isomorphic, so there is only one singleton set, up to isomorphism) represented by the terminal object of the category
- An element of an object A is represented by an arrow from the terminal object to A.
- The the cartesian product of two objects A and B (the set of all ordered pairs of elements (a,b)) is represented by the categorical product of A and B
- The disjoint union of two objects A and B (the set that contains all the elements of A plus all the elements of B) is represented by the categorical coproduct of A and B
- The set of all functions from A to B is represented by the categorical exponential of A and B (notice that the set of all functions from A to {1,2} is isomorphic to the set of all subsets of A)
.....

There is a way to represent EVERY operation in set theory in terms of an universal property ( https://en.wikipedia.org/wiki/Universal_property ) in category theory, and the two representations work exactly in the same way... except for a detail: we cannot distinguish isomorphic sets between each-other. Meaning: we can distinguish them using the language of set theory (meaning: the set { {}, {{}} } is different from the set { {}. {{},{}} } in set theory, but in terms of universal properties, all sets that contain two elements are indistinguishable: you cannot even say how many two-element sets there are.
The arrows, instead, are assumed (by the axiomatic definition of a category) to be all distinguishable from each-other.

So, that's it! Category theory can be used to "represent" the operations used in ZFC set theory (except for this last limitation).

OK, so what for is the "topos"? The point is that WE DON'T WANT TO ALLOW THE USE OF EVERY POSSIBLE UNIVERSAL PROPERTY. We are looking for the MINIMAL SET OF UNIVERSAL PROPERTIES that are enough to be able to represent the operations used in set theory.

Well, it turns out that the minimal set is the following one:
(taken from Wikipedia: https://en.wikipedia.org/wiki/Topos )
"
A topos is a category that has the following two properties:

All limits taken over finite index categories exist.
Every object has a power object. This plays the role of the powerset in set theory.
"
There are a lot of equivalent definitions, but this is the simplest one that I found.
A topos is a category that contains the minimal set of universal properties necessary to encode all the operations required by the language of first order logic (including quantifiers).

Now, the most important point: the derivations that you can produce using this limited set of constructions are not all the derivations of classical logic. And you can build A LOT of different categories with the property of being a topos, by adding different requirements to the basic set of requirements called "topos".

At the same way, you can say that a given mathematical object is a "group" by giving the minimal set of operations and properties that a group must have (you have to be able to take inverses, to form products and to distinguish an element called "unit"), but then there are a lot of different additional requirements that you can add to restrict the set of mathematical objects that match your requirements.

Well, if you want to recover classical logic, you have to use a topos with the additional requirements:
- there are exactly two arrows from the terminal object to the subobject classifier
- there is an initial object (in general, by definition, there is no equivalence of empty set in a topos)
- ... I don't remember now .... Just look at the properties of the category of Sets here: https://ncatlab.org/nlab/show/Set

The category of Sets is the topos that has the additional properties required to make the logic work as the standard classical logic.

And that was only the first question :cry:

I don't think I have time for everything, but we'll see. Sheaves will be for the next time!

P.S. I re-read this and just realized that the correspondence between categories and logic theories that I described is not correct: here's the right correspondence: https://ncatlab.org/nlab/show/internal+logic (ZFC is not even in the list, but higher order logic can be used as an extension of ZFC where even iteration over subsets is allowed, and the corresponding category is called ELEMENTARY topos)

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 than the sequence itself.
In other words, it's all about the quantity of information needed to generate that sequence.
Short sequences are always random.

Yes but I haven't time right now to learn the category theory I'd need. I see a vertical thread of understanding from the idea of a fiber bundle over a manifold, to seeing how that idea generalizes to logic. You've pointed me in that direction several times. So it's not a matter of convincing me that your way is better. The only question is whether you want to explain this to me so I can understand it. I'm pretty close. Tell me the topological space, tell me the map from the open sets to some collection of algebraic structures, that represent propositions and proofs.

Right? A sheaf assigns to each open set of a topolgoical space, a data structure or algebraic object. Tell me the topological space, tell me the map, tell me which structures, represent propositions and proofs. I think that's a specific question we can meet halfway on. — fishfry

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 of the universal set are tuples of elements of our model (the things that we are speaking about: real numbers, for example).
- The subsets are our propositions: they represent the set of all tuples of elements for which the proposition is true (an example here: the proposition "3x + y = 6" is the set of (x,y) such that the equation is true).
- Inclusion between the subsets represents implication.
- Functions are represented in set theory as particular sets of pairs (surely I don't have to explain this to you).
- Relations are sets tuples of elements of our domain.
- There are some distinct points that correspond to the constants of the language.
- The set operations of Intersection, Union and Complement form a Boolean algebra on the subsets of the topology. ( no problem until here, I hope ).

Only in this case, what for is a topology needed? All subsets of this topology are both open and closed. This is a discrete topology (the most particular case of all).

And then the main problem: what about quantifiers? (forall and exists).
You don't want category theory, right? The quantifiers are naturally defined as adjoint functors in category theory, but you said you want only set theory. :roll: so I should reformulate the condition of adjunction of categories in terms of set theory... at first sight it will be a definition that will have to include in itself the algebraic structure of... a category! ( I don't know how to do this :gasp: )
P.S. of course you cannot allow infinite expressions (such as "forall" is an infinite intersection...), since our language is made of strings of symbols.

And then the subobject classifier is the usual set {"true", "false"} plus an evaluation function that for each proposition (subset) returns a value of the set {"true", "false"}

What about sheaves? In this case sheaves are unnecessary too, because we are in a discrete topological space.. I don't know if this program would bring some useful insight really, even if I am able to figure out how to find an algebraic structure on our "topological space" that includes quantifiers :sad:

I need at least the category of sets to be able to include logic, but sheaves are not really related to boolean logic, for what I know.

P.S. I forgot about proofs. Proofs in standard logic are not objects of the model (sets in our case), but it's only a partial order relation between our propositions determined by the rules of logic. Different situation in type theory, where they are represented as objects of our category - meaning: you cannot speak about proofs in standard logic; instead, you can speak about proofs in dependent type theory. And that's why the subobject classifier is not a simple set of values: you have to say not only if a proposition is true or false, but even what is it's proof.

but earlier you didn't remember that you know that, and you were seduced by my example and didn't realize it was inaccurate — fishfry

It wasn't inaccurate, it was a particular case, as you usually do when you give an example..

That's what I mean by mathematical context. You are not being precise enough in your formulations, and that's making it harder for me to latch on to the ideas. — fishfry

Yes, but you can easily find the precise definitions on Wikipedia. I usually don't think in terms of precise definitions.

Re comma categories. Funny story. I checked out a pdf book on categorical logic. In the table of contents they get to comma categories right away. In mathematically oriented category theory books, they get to them much later, and I have read the definition once but didn't understand enough to remember it. — fishfry

That's the advantage of category theory in comparison with set theory: it's more general, but even more simple: it speaks only about the essential features that are needed for proofs to work, and ignore the particular "implementation" (sorry, again a computer-science term). You don't think about real numbers in terms of set-theory, right?

By the way, I saw an Awodey video. I started to read his book but it was too oriented to logic for my taste so I spend more time looking at Leinster. So you see even by inclination I have remained ignorant of categorical logic. But I see a thread. I know what right inverses are and that's basically fibers so I can get this. I just want to understand how that goes to propositions and proofs. — fishfry

OK. I have to go now..

I'd like to understand what you mean when you say Prop is a subobject classifier; given that all I know is that a subobject classifier is {T,F} in elementary logic or defining a subset of a set. — fishfry

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 category (in the topos) is the pullback of the morphism "true" along a unique morphism x:B->Omega. It is isomorphic to a set with two elements in the category of sets (that is a topos), but not in all topoi.

A sheaf, for example is defined only on the open sets of a manifold — fishfry

A sheaf, for example is defined only on the open sets of a manifold — fishfry

You don't need a manifold, you need only a topological space. A manifold is a topological space plus an atlas of continuous maps, right?

I was taking the fibers over the points; but sheaves are defined only over open sets. That's something I'm confused about at the moment. — fishfry

Yes, that's the central point of the whole story: open sets are "more important" than points. In the category of sets an object is "made of" points, but in a generic topos this is not the case. In a topos, a point (of an object X) is an arrow from the terminal object to X. In the category of sets the terminal object is the singleton set and the set of arrows from the terminal object to X is isomorphic to the set of points contained in X, but this is not true in general for a topos.

P.S. I think I know what's the main reason of confusion: you are talking about a sheaf in the category of sets (no category theory needed: only topological spaces defined in set theory), an I am talking about a sheaf defined on a generic topos (category theory needed, set theory irrelevant)

Well, OK, but I don't know which point is the halfway...

Can you give me some references?

OK, I'll drop this topic. Probably nobody is interested... :sad:

OK

Maybe yes. I see sheaves as a comma category, basically.

Can I ask your background again? Now I'm thinking that maybe you learned Coq but don't know the larger context of all these ideas. This is frustrating at my end. I can't tell if you know what you're talking about or if you just read a lot of Wiki pages. That's not meant to be provocative. Only that you are not communicating to me at all. And like I say I know this because I did my mathematical homework last night re sheaf theory. — fishfry

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 about category theory, because it's important for computer science. I am interested in mathematics and physics as an hobby, and I often read new publications from arxiv.org

* What is Prop? Is it the category of propositions? Is it an object in some other category? And in what way is it a subobject classifier, analogous to {T, F} in ordinary logic? — fishfry

Prop is the type of propositions. It's an object of the category. {Prop, true} is the subobject classifier of the topos

* What's Nat? — fishfry

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 from Awodey that you said you have read

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 examples (algebraic structures) of sheaves, that's for sure! I was thinking that we were speaking about the relation between topology and logic. My background is mainly in logic and computer science.