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.

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

I stand by my remark. The tangent bundle of a sphere is most definitely a fiber bundle.

https://en.wikipedia.org/wiki/Tangent_bundle

Can you give me the link you want me to look at? There's been so much back and forth and so many links.

Can you give me the link you want me to look at? There's been so much back and forth and so many links — fishfry

https://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260

Unfortunately, it's not downloadable for free

I stand by my remark. The tangent bundle of a sphere is most definitely a fiber bundle. — fishfry

Yes, of course it is!

Yes, of course it is! — Mephist

That's the intuition I'm working with at the moment, special case that it may be.

Unfortunately, it's not downloadable for free — Mephist

Yes I just checked that out. I'll keep searching around for an insight.

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).
(even the origins of the tangent vector spaces are not in common: they are not seen as subspaces of a common ambient vector space in a higher dimension)

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.

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.

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

Yes! :up:

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! (even if this is not related to the topology of your sets)

Moreover, in this case the topology of the total space (the space made of vector spaces) is "inherited" from the one of the base space: in this sense this is a rather "artificial" example. My example is the most "clean" that I can think of: base space and total space have pre-existing and independent topologies. And it's much simpler than vector spaces: only sets of sets, and functions between sets!

P.S. If you don't like my example because it's made of finite sets, you can "fill the squares" of the total space (it will become a Mobius strip), and connect the points of the base space to make it become a loop! :smile:

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

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.

I'm afraid I share Metaphysician Undercover's misgivings about this remark. — fishfry

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.

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

Yes, and I acknowledged as much.

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

OK, so I have a question: does the number zero exist? Where's the difference between the number zero and the empty set?

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

In category theory sets are described without the making use of the axiom of extension.

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

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.

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

:lol: :rofl: :lol:

Assuming "countable" is what denumerably infinite means — tim wood

That's not what 'countable' means. Here are the definitions:


S is countable iff (S is 1-1 with a natural number or S is 1-1 with N)

S is denumerable iff S is 1-1 with N

S is countably infinite iff (S is countable and S is infinite)


So it's easy to prove that S is countably infinite iff S is denumerable.

So 'countable' does not mean "denumerably infinite", and "denumerably infinite" is redundant, and 'countably infinite' is equivalent to 'denumerable'.

.10101010..., how long is it? How many zeros and ones? As many as there are counting numbers? Or more? ℵo or ℵ1? — tim wood

In this context, for convenience, by 'string' we mean 'denumerable binary sequence'. That said, here we go:

The length of any string = card(N) = aleph_0.

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

Each string in the list has denumerable length. And there are denumerable lists of such strings. But there is no denumerable list of strings such that the list has every possible string in the list.

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

IF the list includes every string, then the diagonal argument doesn't work. But that's not saying much, because we have not shown that there is a list that includes every string. Indeed the diagonal argument proves that there does not exist a list that includes every string. There is no force to an argument that says "If there is a complete list then there is a complete list".

Instead, you start with the question "Is there a complete list?" Then you prove that there is no complete list.


REVIEW of all this:

Each string has denumerable (i.e., countably infinite) length.

The question is "What is the cardinality of the set of all strings?"

If there is a denumerable list that has every string as an entry, then the cardinality of the set of all strings is card(N).

So is there a denumerable list that has every string as an entry? The diagonal argument proves that the answer is No.

And the notation 2^N stands for 'the set of functions from N into {0 1}", which is exactly the set of all strings.

We also prove that the card(2^N) = card(PN).

So the cardinality of the set of all strings = card(2^N) = card(PN).

And the cardinality of the set of all strings does NOT equal card(N).

There are too many incorrect claims in this thread (and forum) to reply to them all. But I'll address one about the axiom of extensionality.

The axiom of extensionality is:

If for all z we have z is in x iff z is in y, then x = y.

That does not contradict the theorem:

There exists a unique x such that for all z we have that z is not in x

and then we have the definition:

the empty set = the x such that for all z we have that z is not in x

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

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.

What about thinking of tangent vectors to a circle in the complex plane rather than a sphere? But I'm not following closely. :sad:

Yes, that's the same kind of function. The point is that you can have a function whose codomain depends on the argument of the function. In type theory this is called a dependently typed function. And that is not only for mathematical functions, but even for functions defined in programming languages ( https://en.wikipedia.org/wiki/Category:Dependently_typed_languages ).

These kind of programming languages can be used as logic languages for mathematics. And a mathematical proof can be expressed as a program in a dependently-typed programming language.

OK, so I have a question: does the number zero exist? Where's the difference between the number zero and the empty set? — 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?

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

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.

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

I was referring to the natural number zero. 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?
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...

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

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?

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

I see that I didn't answer on the main topic here, that was about extensionality.

The fact that "sets are more fundamental than their elements" is true for topos theory, not for topology based on set theory, of course.
In set theory a set is identified by it's elements, and extensionality is an axiom.
In topos theory instead, the "category of sets" (that you can interpret as "the class of all sets") is defined axiomatically as an algebraic structure (a category with some special properties).
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".

P.S. To summarize:
- axioms of group theory ==> axiomatic description of groups
- axioms of category theory + axioms of topos theory ==> axiomatic description of sets and functions (sets are represented objects and functions are represented by arrows)

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

This is exactly why it is contradictory. If there is a set without any elements it is not a set at all. With zero elements the supposed set is non-existent. But if you propose that the number zero is itself an element, such that there can be a set with "zero" as an element, you are saying that there is a set which has an element "zero", but also has zero elements. That is contradictory.

Therefore if we adhere strictly to the method of definition provided, then within set theory, there ought to be no natural number "zero". The natural numbers are defined by the sets which have those elements. There is no such thing as a set which has no elements, this is contradictory, as a collection of things without any things. So zero is excluded as a natural number. by the precepts of set theory. As I explained already, this very same problem was exposed by Aristotle, in slightly different terms, so the Neo-Platonists established "One" as the fundamental Form. To place "zero" as the fundamental form, or "set", is to base the system in contradiction.

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

The symbol has a different meaning depending on how it is used or defined. The natural numbers are used for counting objects. We name a type, apples or oranges etc., and count the number of that type. Since we can name a type and also have no object of that type, we can have zero of that type. "Zero" allows the named type to have meaning, when there is none of them, by allowing that we have the potential for a quantity of that named type, without actually having any of them now. This concept of zero, as the "potential" for objects of a specified type allows us also to count negatives of that type.

Do you see that if we make the number which is named by the symbol, an object itself, then we lose the capacity to use "zero" as the potential for a number of the named type of objects? Zero is itself the object which is named, so there is no such thing as none of those objects. The set of "0" already has an object so it cannot be an empty set.

Try looking at it this way. The natural numbers are used for counting objects. The "count", the number or quantity, is distinct from the objects themselves. "Five", as the quantity of apples on the table, is distinct from the actual objects on the table, it is not a property of the apples. It is only by apprehending the quantity as distinct from the objects, that we can use "zero" as a quantity (natural number). If the quantity of objects was not distinct from the objects, if it were a property of the objects, then there'd be no such thing as zero objects, because there'd be no objects to have that property, "zero". It is only by postulating that the quantity is something distinct from the specified objects that we have the capacity to say that there is zero of the specified type of object. Do you recognize that in set theory, the quantity itself, (as the natural number), is the specified object, therefore we have no capacity to talk about none of those objects? Mentioning that object, as the object talked about, necessitates that there is not zero of that object.

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

It's not a matter of what exists in reality, it's a matter of what is contradictory in principle. To say " I am going to talk about this object, but this object is not really an object, because there is zero of them", is blatant contradiction. To bring this expression out of contradiction we must amend it. I might say for instance, "I am going to talk about a type of object, of which there are none", or I might say "I am going to talk about a quantity, and this quantity is zero". But if I make the category mistake of conflating these two options to say "I am going to talk about this quantity, zero, as an object itself, and assert that there is none of these objects", then I contradict myself.

In set theory a set is identified by it's elements, and extensionality is an axiom. — Mephist

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.

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

So consider that we have a defined property. A "group" is all the members which have that property. We can establish relation between individual members of groups, based on the different groups that they are in. However, the properties are properties of the members, they are not "properties of groups", that would be a composition fallacy. So we cannot proceed toward establishing relations between groups this way, that would be a relation based in a fallacy. Suppose one property is "red", and the other is "hot", and we find that many red things are also hot things, it would be invalid to establish a relation between the property "red", and the property "hot", in this way.

See, "the set", or "group", is based in the defined property. To deal with "the group" as if it were a whole, an object, means that we are dealing with types, the defined property, a universal, rather than the individuals of the group. As an object, the universal, or type is a Platonic Form. A Form, as a universal, is completely different from a particular, an individual. The rules for relating universals to each other are completely different from the rules for relating individuals, because we relate individuals by determining their properties, but a universal is nothing other than a property already.

So here's an example of how we relate Forms or universals. In the Aristotelean way, the more general is "within" the less general, as an essential property, by definition. For example, "animal" is within "human being", as an essential property, like "polygon" is within "triangle", by definition. Further, "human being" is within "Socrates", Socrates being a specific human being. The particular, being the specific thing referred to, the individual human being who bears the name Socrates, is not within anything, and so is called primary substance. Do you see that it is possible for something to be within nothing (within no set), as the more general is always within the less general, so the most specific is not within anything, as primary substance? But that which is within nothing is still something. Now, at the other extreme is the most general, that which is within everything. Never is there the possibility of a set, or defined property which has nothing within it. So we might ask what it means for something to not be within a set, but it makes no sense to talk about a set which has nothing within it.

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

Ah. Thank you. That was very interesting and helpful.

some argue that the real numbers are not truly continuous, — aletheist

Who argues that, exactly, besides the Peirceans on this forum? I've actually never run across this point of view except for here. And how does that square with the intuitionist continuum, which has even fewer points than the standard reals? They can't all be right.

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

LOL. @Mephist was making the point that one can do "set theory without elements" as in Lawvere's elementary theory of the category of sets, which unfortunately doesn't have a Wiki page. But basicaly you can do most of set theory in a purely categorical way. As I understand it you get a slightly weaker version of set theory than the standard theory.

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?

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..

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

Ok you're right. Closets and empty grocery bags aren't really on point, even though they can be helpful visualizations, such as a grocery bag containing an empty grocery bag to visualize $\{\emptyset \}$.

So how about an axiomatic approach? The axiom schema of specification says that if $P$ s a unary predicate, and $X$ is a set, then $\{x \in X : P(x) \}$ is a set.

Consider the unary predicate $x \neq x$. Let $X$ be any set whatsoever, say the natural numbers or the real numbers or whatever set you might happen to believe in. Then we can define

$\emptyset = \{x \in X : x \neq x \}$.

So if you believe in the existence of any set at all, and you accept the axiom schema of specification, then you must accept the mathematical existence of the empty set.

What say you?

That's the problem, I don't believe in the existence of any set. 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.

That's the problem, I don't believe in the existence of any set. — Metaphysician Undercover

I've previously called your philosophy mathematical nihilism, and once again you confirm it. You start by saying you don't believe in the empty set; but it doesn't take long to get you to agree that you don't believe in the existence of any sets at all.

If you don't believe in sets, why go to the trouble of explaining why you don't believe in the empty set? I wonder if that shows that you haven't thought your idea through. Why bother to argue about the lack of elements, when you don't even believe in sets that are chock-full of elements?

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

But this is a strawman argument, "... giving the impression of refuting an opponent's argument, while actually refuting an argument that was not presented by that opponent."

Nobody has claimed sets have "real" existence, whatever that is. Sets have mathematical existence, and that's the only claim I'm making.

I could easily take you down the rabbit hole of your own words. Is an electron "real?" How about a quark? How about a string? How about a loop? And for that matter, how about a brick? Are there bricks? When we closely examine a brick we see a chemical compound made of molecules, which are made of atoms, which contain protons, neutrons, and electrons, which themselves are nothing more than probability waves smeared across the universe.

Do you believe in the existence of bricks? Physics tells us that even bricks are nothing more than probability waves smeared across the universe. We see a brick in its location simply because that's the most likely location for it to be found. Once in a long while, a brick appears someplace else where it has a low probability of being found. I hope you know that this is standard doctrine of modern physics. Do you deny science along with math?

You are painting yourself into an ontological box. Not for the first time, I might add.