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. — Mephist
That's the intuition I'm working with at the moment, special case that it may be. — fishfry
In my opinion, the misleading part of that example is that the tangent planes seem to have some points in common, since they are immersed in an ambient 3-dimensional space. That's not true! The tangent vector spaces are completely separated from each-other (no points in common). — Mephist
Oh I see. Good point. Funny but it never occurred to me to be confused by that. The tangent planes are conceptual thingies attached to each point but they don't "intersect in 3-space" at all. The technical condition is that the total space is the disjoint union of the fibers. I suppose I like this example because it's nice and concrete. For example a vector field is a choice of a single vector from each fiber. So if we have a vector at each point of a sphere that gives the wind direction and velocity at that point, that's a section of a fiber bundle. In set-theoretic terms a section is a right inverse of the projection map. That's how I think about all this. — fishfry
In set theory class many moons ago I proved that "every surjection has a right inverse" is equivalent to the axiom of choice. That makes sense because it says we can always make a simultaneous choice of a tangent vector from each tangent plane. When I found out that a section is what differential geometers call a right inverse, I was enlightened. — fishfry
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. — Mephist
I'm afraid I share Metaphysician Undercover's misgivings about this remark. — fishfry
Yes, and I acknowledged as much.The standard mathematical view is that "the continuum," "the real line," and "the set of real numbers" are synonymous. Philosophical considerations do not alter the conventional mathematical meanings. — fishfry
This was Cantor's view, which is fairly standard among mathematicians today. However, there is a power set for the real numbers, and a power set for that power set, and so on ad infinitum. That being the case, some argue that the real numbers are not truly continuous, despite comprising what is conventionally called the analytical continuum. — aletheist
The problem is that your demonstration, through this technique, produces a misunderstanding of the theory, rather than an understanding. So the criticism is of your technique. You describe topology through reference to set theory, but to understand set theory requires an understanding of extensionality. You demonstrate a misunderstanding of extensionality. The fundamental assumption that a set has extension negates the possibility of an empty set. Therefore your demonstration, which places the set as more fundamental than its elements, implying an empty set, is a demonstration of misunderstanding. — Metaphysician Undercover
The axiom of extension dictates that a set's identity is established by its elements. Therefore a set without elements can have no identity as "a set", and is therefore not a set. — Metaphysician Undercover
Some set-theorists are wont to obscure this fact by saying that the empty set is unique, when in reality it is distinct from all other sets because it is not a set at all; it has no extension. As I explained, this problem was overcome thousands of years ago by making "One" the fundamental "unique set", as the term "unique" implies. — Metaphysician Undercover
When you and I agree on something, that's really something to be afraid of; better move the hands on the doomsday clock. — Metaphysician Undercover
Assuming "countable" is what denumerably infinite means — tim wood
.10101010..., how long is it? How many zeros and ones? As many as there are counting numbers? Or more? ℵo or ℵ1? — tim wood
I'm thinking the number of digits must be countable. And I'm thinking my listing, then, being ordered, is also countable. It's all countable. But clearly that's not correct. — tim wood
if the list is denumerable and complete (just as N is denumerable and complete), then the diagonal argument seems not to work, because any new number generated by the diagonal process will already be somewhere on the list. — tim wood
So if we have a vector at each point of a sphere that gives the wind direction and velocity at that point, that's a section of a fiber bundle. In set-theoretic terms a section is a right inverse of the projection map. That's how I think about all this. — fishfry
OK, so I have a question: does the number zero exist? Where's the difference between the number zero and the empty set? — Mephist
Then I think you should like topos theory: in a topos the object that represents the empty set (the initial object) is not in general required to exist. You can assume it's existence, but it's not required by the definition of a topos. — Mephist
This is a symbol, "0", or "zero". As you seem to be fairly well educated in mathematics, you'll know that it means different things in different contexts. Despite your claim that mathematical languages are very "formal", there is significant ambiguity concerning the definition of "zero". Do you agree that when we refer to "zero" as an existing thing, a number, like in "the number 0", it means a point of division between positive and negative integers? How is this even remotely similar to what "the empty set" means? — Metaphysician Undercover
It's not "the object which represents the empty set" which I am concerned about, it is "the empty set" itself which bothers me. It is a self-contradicting concept. If a set is to be something, an object, then, as an object, it cannot be empty because then it would be nothing. You would have an object, a set, which is at the same time not an object because it's composed of nothing.
So there is a distinction to be made between the definition of the set, "the set of...", and the actual set, or group of those things. If there is none of those defined things, then there is no group, or set of those things, such a defined set is non-existent. There is none of the describe things and therefore no set of those things. There is a defined set, "the set of..." which refers to nothing, no things. It is not an empty set, it is a non-existent set. Only through the category mistake of making the defined set ("the set of..."), into the actual set, can you say that there is this set which is empty. So if we allow that there is this actual set, the set of nothing, then the set becomes something other than the collection of things which forms that set. And we'd have no way to identify any set because the set would not be identified by the things which make it up. — Metaphysician Undercover
The problem is that your demonstration, through this technique, produces a misunderstanding of the theory, rather than an understanding. So the criticism is of your technique. You describe topology through reference to set theory, but to understand set theory requires an understanding of extensionality. You demonstrate a misunderstanding of extensionality. The fundamental assumption that a set has extension negates the possibility of an empty set. Therefore your demonstration, which places the set as more fundamental than its elements, implying an empty set, is a demonstration of misunderstanding.
The axiom of extension dictates that a set's identity is established by its elements. Therefore a set without elements can have no identity as "a set", and is therefore not a set. Some set-theorists are wont to obscure this fact by saying that the empty set is unique, when in reality it is distinct from all other sets because it is not a set at all; it has no extension. As I explained, this problem was overcome thousands of years ago by making "One" the fundamental "unique set", as the term "unique" implies. — Metaphysician Undercover
Natural numbers in set theory are defined as sets: the natural number N is a set that contains N elements. If there is no empty set, there is no zero, right? — Mephist
So, you say that zero is not like the other natural numbers (that are sets), but is only a symbol not well defined. I understand this, but then you say - in "the number 0", it means a point of division between positive and negative integers - but what are negative integers then? Aren't they just symbols? Following your reasoning, I would say that only positive natural numbers are real and all other kinds of numbers are just not well-defined symbols. OK, then how should they be defined correctly? I mean: it seems to be a little "restrictive" to throw away all mathematics except from positive natural numbers... — Mephist
OK, I understand! NO EMPTY SET OF REAL THINGS EXISTS IN REALITY. I agree. But the problem remains: how can you define the other mathematical entities except from positive natural numbers? I think you have to allow the use of symbols that are NOT REAL THINGS if you want to do mathematics, don't you agree? — Mephist
In set theory a set is identified by it's elements, and extensionality is an axiom. — Mephist
An analogous thing to "the class of all sets" is for example "the class of all groups" (in the sense of group theory). You don't describe groups by saying what a group is "made of", but only saying what are the properties of groups: how they relate to each other, and not what they are "made of".
The same is true for sets in topos theory: the theory describes how sets relate to each-other, and not what a set is "made of". — Mephist
Actually, this vector field is a good example of a dependently-typed function. The domain of the function is the surface of the sphere, but what is it's codomain? For each point of the sphere the codomain is a different vector space. But all these vector spaces are identical, except for the fact that they are associated to a different base point. This in type theory is called a parametric type: a type that depends on a parameter in an "uniform" way. And the value of the function is the vector representing wind's direction and velocity, that of course vary with the point on the sphere. — Mephist
some argue that the real numbers are not truly continuous, — aletheist
When you and I agree on something, that's really something to be afraid of; better move the hands on the doomsday clock. But I think the appearance of agreement is based in different principles, so there's really nothing to worry about. — Metaphysician Undercover
Do you see that this proposition denies the possibility of an empty set? The empty set has no identity as a set, and therefore cannot be a set. — Metaphysician Undercover
A closet is an enclosed space in which I hang my clothing.
One day I remove all the clothing from my closet.
Do I still have a closet?
Do I not in fact have a perfectly empty closet? — fishfry
How is that relevant? As Mephist said, a set is identified by its elements. That's the reason why an empty set makes no sense. Clearly a closet is not identified by its elements.. — Metaphysician Undercover
That's the problem, I don't believe in the existence of any set. — Metaphysician Undercover
That any set has real existence has not yet been demonstrated to me. And axioms which allow for the demonstrably contradictory "empty set" lead me away from believing that sets could be anything real. — Metaphysician Undercover
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