I came to this forum about 6 months ago because I wanted to present some ideas that in my opinion are original but are not completely formalized (until now, maybe they will be one day). I discovered that nobody understood what I meant. So probably that's the wrong approach. I am not interested in discussing ZFC. I don't know ZFC and I am not really interested in studying it.

I think I'll have to complete my work in a more formal way and present it to a math forum.

Sorry for bringing trouble. You can continue without me.

First let me put this in context. You said the empty set doesn't exist. — fishfry

I'll not discuss about the empty set any more. Yes, you are right. The empty set exists. You win!

Here's a formal proof in Coq that the Calculus of Constructions is sound: http://www.lix.polytechnique.fr/~barras/publi/coqincoq.pdf

One possibility, seemingly proposed by Mephist (if I understood your responses to MU anyway) is that you put in the empty set as a proper class primitive into the theory, so that (1a) is denied but (1) and (2) are still true — fdrake

Yes, that's correct. You can consider a "topos" as a generalized class of all sets. So, the sets are the objects of the category. The final object of the category corresponds to the set that contains only one element (singleton), and the initial object of the category corresponds to the empty set.
In the general definition of a "topos", the existence of an initial object is not required (because in the definition you want to have largest possible generalization of the category of sets), but the "normal" examples of topoi (such as the category of sheaves, or the category of sets - that are all particular examples of a topos) all have an initial object (corresponding to the empty set).

You get something really similar to that with any mapping t(k):D→Mt(k):D→M, where MM is some manifold in which xx is a point. With the constraint that t(0)=xt(0)=x, the collection of all such maps forms a module ("vector space with elements from a ring"). It's an infinitesimal tangent space attached to the point. It might mutilate intuitions of the real number line, but that doesn't matter, as it seems designed to simplify language and proofs about smooth functions. Whether it's "wrong" or not is just a question of taste and application. — fdrake

Yes, exactly. The intuition of the real number line, in my opinion, is not mutilated but simply different: you have a base space of points that can be built in type theory as limits of convergent sequences of rational numbers. Each of these points is "covered" by an infinitesimal open set.
The weird thing at first sight is that there are no closed intervals such as [0,1], and all functions have to be continuous. But if you think of a function as a model of a physical process, in my opinion that's the ideal mathematical object. In physics, there aren't really discontinuous processes, even in principle, since infinite precision is not measurable. A discontinuous function of time, for example, would mean an infinite velocity of some kind of process.

As I'm sure you know, if a theory's consistent and has arithmetic, it can't prove its own consistency. You always have to go outside a theory to prove that theory's consistency; so consistency of system X is always just relative consistency to some other system Y. If you want to find a model of some axiom system, you need to construct the model through other rules (even if they're incredibly similar). — fdrake

Yes, that's true. But the proof of consistency of a dependent type theory (such as for example Calculus of Constructions) is simply a proof that a given class of programs always terminates and is "strongly normalizing" (meaning: any two programs can always be compared with each-other by reducing them to a normal form, and then checking if they are syntactically equal). See for example https://prosecco.gforge.inria.fr/personal/hritcu/temp/snforcc.pdf.

In reality, in CC the induction principle P(0) and ( forall n:Nat, P(n) ==> P(n +1) ) ==> ( forall n:Nat, P(n) ), and then the definition of the type of natural numbers, is not an internal part of the logic, but is assumed as an axiom. So, strictly speaking, the logic of CC does not include natural numbers. But you only have to "trust" the principle of induction, not the existence of abstract infinite sets.

I'd've thought you'd be quite happy with small categories? :brow: Aren't the models of intuitionistic logic Heyting algebras (from earlier) anyway? — fdrake

Heyting algebras represent only the propositional part of intuitionistic logic, not including variables and quantifiers.

They're sets, or categories which are represent-able as sets. So I'm reading this like: "I don't like sets because I don't like the structures that establish ZFC has models. But I like intuitionist dependent type theory because it has a model! (Which is a collection of sets.)" — fdrake

Yea, this seems not to make much sense.
OK, I'll try to explain.

1. To speak about the elements of an arbitrary set, you need only first order logic. You don't need a set theory. Meaning: first order logic (the rules and axiom of the logic) define the meaning of "forall" and "exists" quantifiers. For example, you can axiomatize a generic group, or a generic ring, using only first order logic. No need of ZFC. ZFC is needed to speak about the relations between sets (inclusion, intersection, subsets, etc...), or the relations between groups, or rings.
2. In principle, the definition of a category requires the existence of two sets: Objects and Arrows. So, two sets, not one.. However, the definition can be reformulated to use only one set (the set of arrows), and the objects can be identified as the identity arrows. So, in principle a generic category can be defined making use only of first order logic, without any set theory.
The problem is that the most important theorems and constructions of category theory (such as functors, natural transformations, Yoneda lemma, etc..) are related to the category of sets. And the category of sets is "the category that has as objects (all possible) sets and as arrows (all possible) functions between these sets". In other words, the category of sets is not defined in terms of category theory itself (meaning: in terms of objects and arrows, as in the definition of a topos, for examle), but is defined in terms of an underlying set theory (that is what it's called a "concrete" category - https://en.wikipedia.org/wiki/Concrete_category).
If you suppose to have ZFC as an underlying theory, I believe (not completely sure) that there would be a problem with this formal definition: you can't say that the objects of the category of sets are the set of all sets, because there is no such set in ZFC. So, if you want to avoid this kind of problems you should use an underlying set theory that assumes the existence of a class of all sets.
Or, for example, you can build category theory based on type theory. Then, the category of sets in this case will have as objects all the types of a given "universe" of types, and as arrows all computable functions. Most of the results of category theory are independent from the preexistent "set theory" that you use to build the category of all sets, but not all of the results.
I don't know the details of these differences, but from a foundational point of view, category theory is not an univocally defined theory. There are even several ways of defining a category for a fixed intuitionistic type theory, because you can "build" the types of arrows and objects in more than one way.
Fortunately, most of the results (practically all of that I know) are independent of the underlying set (or type) theory that you use to define the category of sets. So, if you have a "real numbers object" in category theory you don't have to worry about the underlying logic that you use: you can choose the easiest one to reason about; one where all functions are computable, for example.

OK. Sorry, I give up...

:sad: But that was about ZFC...
Thanks for the answer, anyway!

For Mephist's part, he read a book on category theory but knows very little actual math — fishfry

I would like to hear the opinion of other real mathematicians about what I wrote. For example @jgill or anybody else that can be surely qualified as a mathematician. Could you please point out what I said that is not correct? (this, or even one of my previous posts).

It's perfectly possible (and probable) that I wrote something wrong, but I would like to know what's the mistake that I made.

Was that a yes or a no? Stop dancing. You're wrong on the facts, wrong on the math. Why are you trying to placate Metaphysician Undercover's nutty ideas? — fishfry

I don't understand what I am wrong about. I said there is no proof that ZFC is inconsistent (meaning: nobody has never derived a contradiction from ZFC's axioms), but there is even no proof that ZFC is consistent. That's why I prefer type theory to ZFC. Type theory is weaker but is provably consistent.

Can you show me what I said wrong?

What do you think a set is, if not anything that obeys the rules of set theory? — fishfry

I think the sets that are defined in ZFC are a hierarchical tree-like structure that can be used to model the relation "belongs to" at the same way as the leaves of a tree "belong to" it's root. It lacks symmetry and is too complex. I think in the future it will be substituted by a more elegant and simpler definition. I think it does not correspond to anything in the physical world, so basically yes: it's just an imaginary gadget that obeys the rules of set theory, ad it could be substituted by other similar gadgets that logically equivalent to it.

The fact that he's confused about the empty set, even when shown its existence proof from the axioms of set theory — fishfry

Can you show me a proof of consistency of ZFC set theory that doesn't make use of another even more complex and convoluted set theory?

OK, I see it's not so easy to finish this discussion about the empty set... :meh:

I didn't change idea: there is no contradiction in the axiomatic definition of sets given by ZFC, at least for what has been discovered until now. It has not even been proved that ZFC is not contradictory, however; but since nobody has found any contradiction in ZFC after 100 years of using it, I would guess that it is consistent.
By the way, dependent type theory - at least a subset of the version used in coq - has been proved to be consistent (but of course it is not complete - no way to avoid Godel's incompleteness theorem).

However, I don't see any problem in the definition of an empty set, and the fact that the name "set" could suggest that it has to be composed of at least some elements is not a problem for me, since this is just a name, and names have no role in a formal logic system. If it was called "asodifj" nothing would change, except that it would be more difficult to remember this absurd name.

Mephist seems to have no rebuttal to the arguments which demonstrate that the "empty set" is a contradictory concept, and unlike you, seems about ready to face the reality of this. — Metaphysician Undercover

I understand that your objection is more about the philosophical interpretation of the idea of "set". This in my opinion is not about mathematics, and to say the truth I don't really see the point of this kind of issues. In my opinion, there are only two points of view:
- mathematics, that don't care about the "real" meaning at all
- physics, that cares only about the correspondence between symbols and results of physical experiments.
Anything that has no direct correspondence with the results with physical objects or results of experiments (such as the sets of ZFC, that can be infinite) is an useful mathematical entity, but it doesn't need to have any meaning at all: it's just an useful abstraction.

By the way, if you consider only finite sets, I don't see any problem at all with the obvious interpretation of sets as physical containers of something (that can be even empty).

What I object to is the claim of "existence" for objects which have a contradictory description. — Metaphysician Undercover

I think the word contradictory is not the right term in mathematics. The right term should be "inconsistent", and it has a very precise meaning in of formal logic system. Your "proof" of inconsistency, as I just said before, is not something that contemporary mathematics would accept as valid. Maybe it's valid from a philosophical point of view, but I don't fill qualified enough as a philosopher to discuss about it.

However, I really have no other new ideas how to object to @Metaphysician Undercover arguments, and I don't see the point in repeating continuously the same things...

Does the smiley mean that you don't actually believe what you wrote but that talking to Metaphysician Undercover has caused you to lose your grip? What does the smiley mean? Why did you claim there is no empty set? If you so claim, what do you do with the brief existence proof I just gave? — fishfry

Yea, this discussion is going in circles without any hope of a conclusion. I would like to finish discussing about empty sets!

I don't know. Ask @Metaphysician Undercover

This is a "set_or_nothing", not a "set"... :smile:

:sad: I don't know. I have no more ideas how to explain it.

Maybe you are right: sets cannot be empty. So you have to define another thing, named "set_or_nothing", that is a set or it's nothing. Just substitute the word "set" with the word "set_or_nothing" everywhere, and everything will be fine!

Sorry, but I have no more ideas to explain this... I prefer using programs that manipulate symbols without wondering what those symbols really mean :razz:

Excellent reference! (Derek Elkins's response).

I appreciate your efforts to make sense of this for me. I am not just trolling. — Metaphysician Undercover

:smile: good to know. Of course you don't have to believe me as a matter of principle. Usually I make a lot of mistakes when I write.

To define "extension" as a property which something can both have and have not, at the same time, is just a trick of sophistry, designed to dodge application of the law of non-contradiction — Metaphysician Undercover

Yes, of course. A property such as "X has extension" is a boolean value (true or false) associated to X. It cannot have both values at the same time.

I don't see this. I cannot see how you made the contradiction go away. All I see is a trick of sophistry, which hides the contradiction behind the illusion that zero extension is some sort of extension. — Metaphysician Undercover

"null" is an attribute of "extension": an extension can be "null" or "not null". A set can be defined as something having extension (following your definition). null extension => empty set; not null extension => not empty set. Maybe this is a trick of sophistry, but it avoids the contradiction.

So let's look at this example of the segment and the point. You define "point" using "segment". A point is a segment without any length. So the property here is "length". The definition of "length" as you described with "extension" is irrelevant. But from my perspective we need to ensure that "length" is not a sort of property which a thing can both have and not have, at the same time, or else the definition of length would become relevant, as a sophistic trick. The subject, or category is "segment", a point is defined as a type of segment. So we now need a definition of "segment" to make sense of what a point is. Remember that we have just allowed for a segment with no length, so "length" cannot be a defining feature of "segment". How would we proceed to define "segment" now? — Metaphysician Undercover

A segment is defined by giving two points. if the two points are coincident (the same point), then it's the same thing as only one point. If the two points are distinct, the length is the measure of their distance from each-other. I don't see any problem with this definition.

I submit that this is a similar situation to what we have with "set". If we define "empty set", such that it is a real set which has no extension, then "extension" cannot be a defining feature of "set" without allowing that "extension" is a sort of property which defies the law of non-contradiction.. — Metaphysician Undercover

Yes, that's the same thing: you don't define what a set is, but just give some properties that any set should have, and one of the attributes of a set is the fact to be empty or not: this is just an attribute of any set: no need to define the word "extention".

No I don't see any contradiction here. There is nothing to imply that a point is a line without length. That would be contradictory when #2 says that a line is length — Metaphysician Undercover

In my opinion to say "A line is length without breadth" is like saying "A line is a rectangle with zero width". He means that real objects have 3 dimensions, but a line is like a real object that has only 1 dimension. The other two dimensions are missing.

Let's look at it this way. The "line" introduces a new property which the preceding "point" has not, "length". The "surface" introduces a property which the preceding "line" has not, "breadth". But the "surface" also maintains the property of the "line", which is "length". Following this pattern, the "line" ought to maintain the property of the "point". But "no parts" is a sort of negation of a property, instead of a proper property. We can say that this negation is the property which the "point" has. So if we understand "no parts" as a negation of all properties, we'd have to understand the property of the point as "no properties", and this would be contradictory. It would be contradictory, to say that the property of a thing is that it has no properties. However, we do not understand "no parts" as "no properties", so the point is defined by what it does not have, and what it does have is left empty or undefined.

In the case of the "empty set", the property which it does not have is extension. However, being designated as a "set", it also has whatever property is proper to a "set". If extension is a defining property of a set, we have contradiction because we talking about a thing which is said to have extension (by the type of thing that it is said to be), yet it is also said to have no extension by the value given to that property. — Metaphysician Undercover

Well, we could do the same with sets: adding properties instead of subtracting
- an "point-set" is a set with no parts
- a "line-set" is an extension of the "point-set" that introduces a new property: the number it's parts.
For me, that's the same logical construction. Why this should not be allowed?
At the end, they are all similar ways to do the same thing: attach some properties to an object to describe it without giving an explicit definition in terms of other objects!

Yes, however in my opinion Anders Kock's book ( https://users-math.au.dk/~kock/sdg99.pdf ) is not so difficult to understand. d in my opinion should not be interpreted as a number, but as the base of a vector space made of infinitesimal numbers attached to each of the real numbers of a "base" space.

OK, I see:

This implies that if one reads for example xd1 = 8d1 this not necessarily means x = 8. However if
xd = 8d ∀d ⇒ x = 8.

I don't really understand this. For what I understand, d should be treated as a differential operator and 8d is another differential operator. The elements of the real number line are pairs made of an element of a ring and a differential operator: to each element of a ring is attached a linear vector space. Maybe I missed something...

OK, let me look at that link.

All you need to do to turn any non-zero dividing element into a zero dividing element is to multiply it by d. So all elements zero divide. — fdrake

But you cannot multiply by d. You can multiply by (0, d*1), for example, not by d. All non-zero elements are all the elements of the form (x, d*y) where x is not zero.

Yes, but d is not a real number. d is a linear operator (like derivatives). The real numbers are of the form (a, d*b). In this case, for example, (0, d*3) does not have a multiplicative inverse, since it's base part is 0.
d is the base vector of a vector space attached to each number (ring element) on the base space.

You cannot divide by a number that has it's "base" part 0. That works at the same way as the usual real numbers. Can you make a more concrete example?

the construction doesn't satisfy the field axioms — fdrake

Why not? Which of the field axioms are not satisfied?

Moreover, every function from the constructed number line with infinitesimals to itself turns out to be smooth (so, continuous and differentiable). If I've read right anyway. — fdrake

Yes, that's true. All subsets of the real line are open, so all functions are continuous (and differentiable).
Of course it's not equivalent to the "normal" real line, but calculus works at the same way. Why is this a problem?

Hmm... sorry, I didn't even read @aletheist posts :gasp:

OK, now I read it, but I don't quite agree on all that he writes

For example example this part:

A set is a bottom-up conception, assembling a whole from discrete parts. True continuity is a top-down conception, such that the whole is more fundamental than the parts. — aletheist

"true continuity" can be defined even using standard set theory. Actually, even category theory can (and usually is) be based on standard set theory.

The fact that you have to use intuitionistic logic with it's weird rules about double negation, to make sense of "d^2 = 0" even if "d =/= 0" in my opinion is just a "wrong" definition of the negation operator: it should be called "complement of" instead of "not" (for example we should say "d belongs to the complement of 0" instead of "d =/= 0"). The complement of 0 is an open set. The complement of "the complement of 0" is another open set, disjoint from the first one, that includes not only zero, but zero and the infinitesimal neighborhood of zero.

It is true that the whole theory is usually formulated in terms of topos theory, and it seems way too abstract to be used as the "normal" theory of real numbers, but in my opinion it doesn't have to be presented in this way.

Not sure what are R and D in that formula.

In Anders Kock's book ( https://users-math.au.dk/~kock/sdg99.pdf ) D is an infinitesimal interval centered on x = 0 and defined algebrically by x^2 = 0 (see the definition at page 2).
R instead is the base space, defined agebrically simply as a commutative ring (built starting from two fixed points 0 and 1). The "real" real line is made of pairs of elements (a,b) of R (see definition 1.1 at page 3), where "a" is the point from the base space (the finite part of the number) and "d * b" is the fiber over "a" (the infinitesimal part of the number). "d" is an element of D.

"Contradiction" has nothing to do with identifying things with physical objects, it relates to how words are defined. So for example, if "set" is defined as something having extension, and "empty set" is defined as a set having no extension, then there is contradiction here. "Empty set" breaks the rules expressed in the definition of "set", and therefore cannot be a set. — Metaphysician Undercover

OK, let's follow you definition of "set" (that is not the definition used in ZFC set theory, but we are considering an alternative definition because we do not accept MY mathematical rules).

Definition 1: "a set" is something having extension
Definition 2: "an empty set" is "a set" having no extension

Substitute the word "a set" from D1 in D2 and you get P1:
Proposition 1: "an empty set" is something having extension having no extension

P1 could be rewritten as: "an empty set" is something "having extension" AND NOT "having extension"

So we get a contradiction "H and not H" where H is "having extension".
Then, the two definitions D1 and D2 cannot be used at the same time.

Let's follow your reasoning and keep only D1: there is no empty set with this definition.

But now we are not finished yet: we have still to define what is "extension".
I think you have two possibilities:

1. define "extension" in terms of another property (something like "occupation of space"? I don't know..)
2. consider "extension" as an undefined "primitive" notion

- In case of 1. you end up in an infinite chain of definitions (of course you cannot define "extension" in terms of "a set", right?)

- In case of 2. you just did what today's mathematics do, just replacing the primitive notion of "set" with the primitive notion of "extension" and changing the definitions accordingly.

But now what prevents me to consider a "null extension"?
"extension" at this point is an undefined notion, so "null extension" does not generate any contradiction now.
And if you allow "null extension", why not allow "empty set" and consider "set" to be a primitive notion instead?

Of course you can say that the term "null extension" is not allowed (meaning: you are not allowed to use the attribute "null" with the word "extension"). But this is now an arbitrary limitation of the terms (a choice that you made in defining the concept of "extension"), and not a necessary condition to avoid a contradiction.

Following this argument that "null extension" is not allowed, you could say for example that a segment with "null length" is not allowed, so a point is not a segment. That is OK, but it's not due to a contradiction: it's only a choice of your definitions. Defining a point as a segment with no length does not create any contradiction, if you consider a segment as a primitive notion and a point as a derived notion.

Do you agree?
If you don't agree, then try to derive a contradiction due to the introduction of the concept of "an empty set" without making use of other undefined terms, such as "extension".

P.S. Try to take a look at Euclid's elements (https://en.wikipedia.org/wiki/Euclid%27s_Elements)
Here are the definitions, from Book 1 (taken from the book 'The elements of Euclid" by Oliver Byrne)

1. A point is that which has no parts
2. A line is length without breadth
5. A surface is that which has length and breadth only

Are these definitions contradictory?

I understand what you mean. But the word "contradiction" in mathematics has the meaning that I said: "A and not A" is not provable for any A.
What you call "contradiction", the impossibility to identify the terms of the language with physical objects, is not considered as a problem in mathematics: it's simply ignored.

Here's a famous quote from Bertrand Russell about mathematics:

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

(https://www.brainyquote.com/quotes/bertrand_russell_402437)

This is at the same time both a limitation and an advantage: it gives you the freedom to invent new concepts, but you lose the relation between mathematical concepts and the physical world!
This is a choice that mathematicians have done at the beginning of 20th century, mainly (I believe) to get rid of the paradoxes arising from the use of infinity and infinitesimals.

However, in my opinion this is the natural development of Aristotle's logic: the formalization of the rules of deduction. The rules of deduction (used in proofs) should not depend in any way on the meaning (or correspondence to real physical objects) of the words.

So, you say that this is all wrong, because you are allowed to create axioms that don't have any correspondence to reality. That's true. But what is the alternative? After all, these mathematical constructions based on "nothing real" happen to be very useful to build models that agree with experiments.
As far as I know, I think it would be possible to reformulate all mathematics without making use of empty sets at all. But would this make any difference?

If you are merely talking about a set of rules by which symbols are related to each other, then there is no reason why we can't discuss these rules in plain English. I think that your refusal to discuss this in plain English is evidence that you know that there is deception within the system. So, either you discuss these rules in plain English or I level the accusation that you're attempting to hide deception behind your language. — Metaphysician Undercover

Yes, I confess that I am trying to hide a deception behind MY language :rofl:
The things that I wrote can be found in any introductory book to mathematical logic, and they are very clear for most of the people that write on this forum. If you really wanted to understand it, just buy a book and read it!

I don't see how your art analogy works for you. An individual can glance at a painting, and find it ugly without analyzing it, just by apprehending a few prominent features of it. — Metaphysician Undercover

Yes, but you are not even trying to take a glance at the painting! You don't have to be an expert to understand how mathematical logic works. And unlike other parts of math, you don't even need to learn some other more fundamental concepts before.

It's completely unreasonable for you to say that I must accept the system's axioms in order to judge the system's axioms, when acceptance is dependent on judgement, and acceptance precludes the possibility of fair judgement. A conclusion cannot be incompatible with the premise, so if I accept the axioms, it is literally impossible for me to produce a judgement against them. Therefore you are being completely unreasonable. — Metaphysician Undercover

You don't have to accept the axioms. You have to prove that assuming those axioms leads to a contradiction. You have to use the rules of logic to produce a sentence of the form "A and not A" (I am not sure if "true" and "false" are terms of first order logic, maybe I made a mistake before saying that you
have to derive "false"). As @fishfry has explained to you many times, this is like in the game chess: you have to show that, starting from a given position, white can checkmate. If it's not possible to produce a sentence of the form "A and not A", it means that the axioms are consistent (not contradictory).

The interpretation of the terms as sets (and then the meaning of the sentences) is a different issue.
You can argue that the terms that ZFC calls "sets" are not exactly correspondent to what we "intuitively" think to be sets, and a lot of people (even mathematicians) have this kind of objections to ZFC. But this is not about the consistency of the theory; this is about it's "meaning".

You do recognize that "false" and "true" are assigned to the premises, not by what is determined by the logical system, (which is validity), don't you? Inconsistent, or contradictory premises, is not determined by the logic of the system. — Metaphysician Undercover

NO. "false" and "true" in first order logic (the logic used in ZFC) are purely SYNTACTICAL expressions. They are determined ONLY by the logic of the system. That's the way it works!

Look at the definition from wikipedia ( https://en.wikipedia.org/wiki/Consistency ).
"""
although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory T is consistent if there is no formula "phi" such that both "phi" and its negation "not phi" are elements of the set of consequences of T.

The set of axioms A is consistent when "phi" and "not phi" belong to "sentences derivable from A" for no formula "phi".
"""

Then, there is a theorem ( https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem )
"""
that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It makes a close link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proven in particular formal systems.
"""

I criticize the axioms according to how they are expressed in English. If the fundamental axioms could not be expressed in English, or other natural languages, they would be meaningless. Terms need to be defined. — Metaphysician Undercover

In a formal logic system TERMS DON'T NEED TO BE DEFINED. That's why it is called "formal" logic.

But what type of meaning could this be, if when it is represented in English it is contradictory? — Metaphysician Undercover

It can be a meaning that has nothing to do with the English meaning of the words. The proof of the theorem shows that a model always exists (if no contradiction is derivable) because it can be built using the strings of symbols of the formal language itself!
Probably that's the part that you strongly disagree with. But if you want to criticize the proof of Godel's completeness theorem, you should at least read it! That's what I meant by "looking at the paintings" before.

Let's say I have some "True continuity" X. Like a line X=(0,1). Let's say I can take "parts" of it in the above manner; arbitrary subintervals. (0,a), (a-1/n,1) are subintervals for any a less than 1 and greater than 1/n. Since they're parts of a true continuity, the true continuity ensures the existence of their intersection; limit the process of intersection over n and get the limit {a}. If I take the union of all such limits, since a was an arbitrary member of (0,1), it produces (0,1). So starting from a true continuity, like a line segment, you can get discrete numbers, then build up the true continuity out of the individual numbers through a union. It's top down and bottom up at the same time.

Do you agree this process is legitimate? — fdrake

But you can't start from ANY real number "a". If you define real numbers as limits of rational numbers, "a" should be rational, or should be itself a limit of a sequence of rationals. In a constructivist logic you have to define how "a" is "built".

You then stipulate that any function ff defined on D→R∪DD→R∪D has some unique constant bb such that for all d∈Dd∈D f(d)=f(0)+dbf(d)=f(0)+db. There's also a property that d2=0d2=0 for all d∈Dd∈D. This looks like placing a family of infinitely small line segments around every point in R∪DR∪D. For functions, it gives a "tangent space" of a function at a point confined to infinitesimal neighbourhoods around it. It "smears out" the real line (and function values) into something that can't be element-wise disassembled in the same way (if you fix a point of RR, this corresponds to an infinitesimal neighbourhood around that point in R∪DR∪D) — fdrake

Exactly. This is a sheaf of linear tangent spaces built on the base space of Cauchy sequences of rational numbers. Similar to the sheaf of all vector spaces tangent to a sphere.

Well, you've arrived just where I have a problem, or, rather, where I'm unclear and confused, because I'm not arguing against any well-known fact, but rather I seem to be stuck in some misconception or misperception.

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

ℵo. That's the cardinality of infinite DISCRETE sets.
ℵ1 is the cardinality of powersets of sets whose cardinality is ℵo.

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. — Metaphysician Undercover

If you want to prove that ZFC is inconsistent you have to derive "false" using the rules of ZFC's logic. You can't do it using english language, as you are trying to do.
You can't be an art critic without looking at the paintings!

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)

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?

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.

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.

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:

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.

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:

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!

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)

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

Yes, of course it is!

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