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:

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...

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? — Mephist

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.

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. — Mephist

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.

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!) — Mephist

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.

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... — Mephist

ok

without this requirement, a category may even be made of 3 objects and 4 arrows... — Mephist

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

Been following along, maybe this helps.

I guess if we took $\Omega=\{0,1\}$, equipped it with the topology $T=\{\emptyset, \{0\},\{1\},\{0,1\} \}$, with the usual unions and intersections and complements, we could conceive that:

$\{T, \cup,\cap,{}^{c}\}$ is some sort of algebra (where the symbols have their usual meaning). Every open set is also closed.

If we have a proposition $P_i$, an interpretation $I$ is a mapping from $P_i$ to $\{0,1\}$. If we imagine "starting with $\Omega=\{0,1\}$", a proposition is something like pre-image of $I$. If we have a family of such propositions, $Q = \{P_i, i \in J \}$, we could imagine each proposition being such a pre-image. If we consider propositional formulae from this alphabet containing at most $K$ symbols, $\times_K Q$, we could equip this with functions to $\{0,1\}$ (of up to arity $K$) that evaluate to true or false. These are then truth functions. Like $f: Q \times Q \rightarrow \Omega: (x,y) \rightarrow 1$ iff $x=y=1$ else $0$ is the truth function for AND.

If we extended the topology to the product space $\times_{K} \Omega$, 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 $K$ proposition symbols. If we took the intersection of {1} with {1} we get 1, intersection {1} with {0} we get 0... and can construct AND as a truth function on the open sets of this topology.

If we set out the production rules of propositional logic on $Q$, an "open set" might be an interpretation of a syntactically valid formula. There looks to be a topology here: closed under finite conjunctions (intersections), finite disjunctions (unions), interpret the empty string as mapped to empty set.

The negation of a syntactically valid formula is a syntactically valid formula, so the closed sets are all open... Any subformula of a well formed formula is syntactically valid. Well formed formulae are closed under finite conjunction and finite disjunction (since it's a finite alphabet this gives the appropriate topology properties, or somehow corresponds to them). "Pulling back" a truth function along an interpretation might give the propositions which satisfy it. If we "pulled back" a tautology (the truth function which is 1 for all arguments), we would (probably) get a theorem - signified by all the truth table rows being in the pull back. Pullbacks and fibre products are intimately related (somehow, I'm not sure on the details, I've been working through Category Theory for Scientists and pullbacks/fibres were my last session).

This starts looking suspiciously like a correspondence is in play between the algebra of sets on the product topology of $\times_{K} \Omega$, and the production rules on the propositional symbols. The possible "propositional assignments" that satisfy an interpretation maybe float above an interpretation as an algebra (algebraic structure, anyway).

Maybe it doesn't help though, it's very scattered.

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...

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.

and open sets are naturally represented as types in type theory — Mephist

If we continued with the propositional classical logic in my example, and we have an interpretation from $\times_K Q$ (the symbol set product-ed with itself $K$ times) to $\Omega=\{0,1\}$, and if we gave $\Omega$ the discrete topology as I said. If we then stipulated that the interpretation was "continuous" in the sense of topological spaces (the pre-image of every open set is open), would the pre-image of any open set of $\Omega$ with the discrete topology be a type, and thus a proposition? Or a collection of propositions which co-satisfy/are equivalent?

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?

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?) — Mephist

So you get a discrete space of points split in two equivalence classes: true propositions and false propositions. — Mephist

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 $\times_K \Omega$ through the interpretation $I$ made sense to you.

Something like, if we have a logical connective's truth function $f$ from $\times_K Q \rightarrow \Omega$, and we glue together open sets on $\times_K \Omega$ 's discrete topology through the fibre $\{ I(a)=1=f(c) | a \in \times_K Q , 1 \in \Omega, c \in \times_K \Omega \}$. I guess inducing a topology by pulling back a topology through a continuous function and a connective (dunno if that works at all).

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

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

I'm not sure I'm seeing that yet. I'm working on getting a bottom-up understanding of fiber bundles and this will take a couple of days or more for me to sort out a coherent argument. My general idea is to start with the Cartesian product of sets as an example of the fiber bundle idea; then work through the definition of a manifold, which (ignoring all the technical details) is an example of a Cartesian product. From there to sheaves is easy.

Now having laid out carefully a bottom-up understanding of fiber bundles, I'm going to want you to be very specific in putting your ideas in this context. I may or may not be successful in pulling this together because my knowledge of manifolds is weak but I think adequate to the task once I review some things.

Meanwhile I'll probably stay out of this for a while.

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

Honestly there's much less here than meets my eye, at least, from the standpoint of knowing what a fiber bundle is in topology. You are not making any connection with the fundamental definitions and you're not supplying the details. My wild-assed guess is that your Coq teacher mumbled something about fiber bundles and you haven't thought the details through. Is that uncharitable? If not, and you do know how to drill your idea down to the definition of a fiber bundle, let's just say I wish you'd work harder on exposition. Hopefully I'll pull my idea together and that will give us something specific to work with.

would the pre-image of any open set of ΩΩ with the discrete topology be a type, and thus a proposition? Or a collection of propositions which co-satisfy/are equivalent? — fdrake

EXACTLY a question I put to @Mephist the other night. That's identical to my understanding of what's being proposed, but I don't entirely believe it till I work through fiber bundles from the bottom up, which I'm working on. I'm gratified to find agreement on our interpretation of this idea.

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. — Mephist

Oh you DO know this material. It must just be your exposition that I can't understand. I'm happy to see this paragraph. I understand it, I agree with it, and I'm hopeful you'll frame the logic argument in terms like this. Maybe you already have. I'll work at this some more.

This starts looking suspiciously like a correspondence is in play between the algebra of sets on the product topology of ×KΩ×KΩ, and the production rules on the propositional symbols. The possible "propositional assignments" that satisfy an interpretation maybe float above an interpretation as an algebra (algebraic structure, anyway).

Maybe it doesn't help though, it's very scattered. — fdrake

I haven't worked through this yet but it looks very promising.

Bleh, constructing the connection from the set theoretic side is extremely indirect. I think there're three components.

(1) Showing that the open sets of a topology are a Heyting algebra (or associate with one).
(1a) This should obey some restriction property, so that if we restrict the topology to a subspace, it will also be a Heyting algebra. This should follow from (1) directly, as if all open sets in any topology form a Heyting algebra, then so too will the subspace topology induced through a restriction. This probably turns out to let us take subsets of a language, like individual well formed formulae, and give them models in exactly the same way. Such a restriction would probably be continuous, as if we endow the subspace with the subspace topology and treat the mapping between them as an indicator function, the pre-image of every open set would be open just by definition. So the "continuity" of the interpretation looks to be a by product of (1) and the continuity of the inclusion map from a space with a topology to a subspace which is the image of that map with the subspace topology. The restriction map is probably quite similar to a subobject classifier, since it is a characteristic function for the set being used in the restriction. The overall intuition I have is that if we have a model, we can go to a larger model by "pushing back" along the restriction map.

(2) Showing that a model of intuitionist logic is a Heyting algebra (or associates with one)
(2a) The ordering relation in the Heyting algebra respects syntactic entailment, since we're using a Heyting algebra as a model, the logic is complete, it also respects semantic entailment. If it "respects semantic entailment", it should "respect the mapping from the logic to the Heyting algbra", because that's how we provide the formulae with models.

(3) (1) and (2) together give us that the open sets of a topology are a model of intuitionist logic. The interpretation function would then be taking the well formed formulae of the logic and associating them with their corresponding open sets, which are isomorphic to set "domains" that satisfy the formula, and have their own Heyting algebra.

The category theoretic construction lets you come at this some other way. If we re-parse everything in category theoretic terms the same things would hold. But it is probably true that it holds more generally for topoi that aren't representable as sets.

Edit, trying to translate between @Mephist's intuitions and our stodgy old analysis ones: if we drew out a category-theoretic diagram that had intuitionistic logic models on it as a category, then topological spaces as a category, then the syntax of intuitionist logics as some other object, we could construct the arrows:

Intuitionistic logic syntax ---interpretation>> intuitionistic logic models
Topological spaces ---replacement of union and intersection by join and meet and other stuff>> intuitionistic logic models
We could construct a model of an intuitionistic logic through a pullback of the interpretation and the other map? So that the "fibres" (pullback elements) are the open sets which model our specified syntactical objects. The topological space would model the whole logic when and only when the whole space is in the fibre product, maybe anyway.


There's a comment that goes through some of the construction here.

Still unclear, but I'm starting to see how it could be the case.

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've lost track of this lengthy discussion and how it relates to the topic of the thread. Can one of you describe in layman's terms what you are attempting to do? Even though I'm a retired mathematician it's mostly beyond me.

If a simple explanation is not possible, say so. That's OK :chin:

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:

Beyond my meager knowledge of topology. I'm more a metric space guy. I hope others reading this material can follow it. Is there value in showing the same algebraic structure? Like then using results in one to prove results in the other?

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.

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?

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. — Mephist

Yes, I'm familiar with the notion, although I have no experience in an algebraic venue. For instance, many years ago 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.

I also showed that the traditional Tannery's theorem makes far more sense when embodied in more general infinite compositions rather than merely series and products, or integrals. Not quite what you are stating, but close. You guys are on a roll! :cool:

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... )

analytical continuation of a complex function has something to do with chaotic systems? Did I understand correctly? — Mephist

Dynamical systems! Actually, I was more general, showing certain sequences of linear fractional transformations can have their regions of convergence expanded by the use of fixed points. Continued fractions are a special case.

Here is a simple example illustrating the use of a fixed point:

Given $F(z)=\frac{1}{1-z}$

Then $F(z)=1+z+{{z}^{2}}+\cdots ,\text{ }\left| z \right|\lt1$

For $\beta=0$ the T-fraction expansion of the power series is

${{T}_{n}}(\beta )=1+\frac{z}{1-2z+\frac{z}{1-z+\frac{z}{1-z+\ldots +\frac{z}{1-z+\beta }}}}$

The continued fraction value is by definition

${{T}_{n}}(0)\to CF$


Here it is found that

${{T}_{n}}(0)\to F(z),\text{ }\left| z \right|\lt1$ and

${{T}_{n}}(0)\to \tfrac{1}{2},\text{ }\left| z \right|\gt1$

Whereas

${{T}_{n}}(\beta )\to F(z),\text{ }\left| z \right|\gt1,\text{ }\beta =z$

Here
$\beta =z$ is the repelling fixed point of the function $t(\zeta)=\frac{z}{1-z+\zeta }$

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

Exactly not what you said the other day, when you started out by saying that the underlying set consisted of all n-tuples of real numbers then chose not to respond to any of my questions. That's why I"m trying to get you to nail down your definition. If you say you have a fiber bundle I'm entitled to ask what is the base set, what is the total space, what is the map? It's a perfectly sensible question.

Which model? You haven't said anything about models.

Is your previous claim now retracted and you are now making this different claim? The one with the clock?

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.

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. — Mephist

I'm sorry to hear of your personal health issues.

Have you ever heard it said that if you can't explain something clearly, you don't actually understand it? That's the sense I get from your posts. I could be wrong. Hope your health issues turn out well.