Traffic laws are a made-up game too. — fishfry

Uh, no, laws are not games. What kind of game do you get penalized for not playing?

But the word "contradiction" in mathematics has the meaning that I said: "A and not A" is not provable for any A. — Mephist

But I haven't yet accepted your mathematical rules. I am judging them as to whether or not they ought to be accepted. So I can only judge "contradiction" according to what it means in English.

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. — Mephist

"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.

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. — Mephist

"Meaning" is not necessarily dependent on correspondence with physical objects, it might be derived from relations within a conceptual structure. That's why I outlined two distinct types of "consistency", consistency within a particular structure, and "consistency" in how that structure relates to outside principles. Notice there is no necessity for correspondence with physical objects. But when correspondence with physical objects (what some call "truth") is one of those outside principles, then the conceptual structure might be judged in relation to this principle.

So, you say that this is all wrong, because you are allowed to create axioms that don't have any correspondence to reality. — Mephist

That's not what I'm saying is wrong. What's bad is if there is contradicting axioms, like in my example above. Suppose people are creating axioms, and the axioms are not necessarily corresponding with reality. There's no inherent problem with that. Now suppose a problem in application of the axioms appears, possibly because the axioms don't correspond with reality, perhaps some sort of paradox appears or something when people try to apply the axioms. So the people creating axioms decide that if they change this axiom, or create another axiom, the problem can be avoided. But maybe they don't realize that the new axiom contradicts another axiom, or if they do, they might still be inclined to accept it because it makes that particular problem go away. However, I think the contradictory axioms are bound to create other problems further down the road.

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

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). — Mephist

I recognize that is not the proper definition, I wrote something simple as an example.

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. — Mephist

I agree that the definition of "extension" is in principle irrelevant. But no matter how "extension" is defined, it doesn't resolve the contradiction which is involved with something that, at the same time, both has and has not extension. 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. If this is the case, then the definition becomes relevant.

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? — Mephist

Do you not apprehend the trick of sophistry here? The law of non-contradiction says that the thing cannot be categorized as both "having extension" and "not having extension". Now, you introduce "null extension" as if it allows that the thing to be categorized as "having extension", as if "null extension" is a sort of extension, when "null extension" really means "not having extension".

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. — Mephist

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. But it cannot be, because if zero extension was some extension it would not be zero. I do understand that it is a matter of definition, but I do not see how defining a property, whatever that property is, extension, length, or whatever, in such a way so that a thing can be said to both have that property and not have that property, at the same time, is anything more than a trick of sophistry designed to circumvent the law of non-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?

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..

[q

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? — Mephist

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

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. — Mephist

Still sets tho. $x+db \in R \cup D$.

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? — Mephist

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.

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.

I'm only pointing out that if sets can't provide a model for @aletheist 's intuition of continuity, since they consist of distinct entities (you can distinguish infinitesimals from each other, and make claims like $x+db$ lays in some set), then neither can the synthetic axioms presented in that link, as they concern sets.

(Though, they can be presented in a more general context. What I'm pointing out is that while sets aren't necessary to talk about continuity in that sense, they don't preclude it by themselves either.)

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.

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"). — Mephist

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

It's not really about the reals and analysis as usually thought of, the construction doesn't satisfy the field axioms. 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.

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?

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

Multiplicative inverses, you have zero divisors from the infinitesimals and that blocks it.

Why is this a problem? — Mephist

It's not, it's just a point of difference. Having a number system that's built to embed differential geometry intuitions and make differential equation diagrams rigorous is very cool.

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?

You cannot divide by a number that has it's "base" part 0. That works at the same way as the usual real numbers — Mephist

Ok. Assume d had a multiplicative inverse. Also d^2=0. Then d.d^-1=0.d^-1. Which gives d=0. But d is nonzero. So d does not have a multiplicative inverse. All nonzero elements need a multiplicative inverse for it to be a field.

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.

The real numbers — Mephist

The real numbers $\mathbb{R}$ are a field (without 0). The real numbers augmented with the set of numbers defined in the previous post's link are not.

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. If you quotient out the infinitesimals by mapping to standard parts it's a field again, as it's just the reals. Zero divisors can't have multiplicative inverses.

If you have two nonzero ring elements a and b, and a.b = 0, assume wlog that b has an inverse, then a.b.b^-1 = 0.b^-1, then a=0, so b has no inverse.

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.

Page 4. Properties.

"However if xd = 8d for all d then x=8". In that link, they are multiplying by d.

OK, let me look at that link.

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...

I don't really understand this. — Mephist

It seems to let you rigorously think of dx as an infinitely small translation/length with the expected connections to calculus and differential geometry.

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.

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!

"true continuity" can be defined even using standard set theory. — Mephist

No, the whole point of talking about "true continuity" is to distinguish it from (analytical) "continuity" as defined in accordance with standard set theory. The real numbers do not possess true continuity, because numbers of any kind are intrinsically discrete. However, they serve as a useful model of continuity, adequate for most mathematical and practical purposes.

Actually, even category theory can (and usually is) be based on standard set theory. — Mephist

I am not a mathematician, but my understanding is that this is exactly backwards. Set theory can be established within category theory, but category theory cannot be established within set theory. "Set" is one of the categories, but there are others that need not and do not conform to standard set theory.

I am not a mathematician, but my understanding is that this is exactly backwards. Set theory can be established within category theory, but category theory cannot be established within set theory. "Set" is one of the categories, but there are others that need not and do not conform to standard set theory. — aletheist

I'm not personally in accord with this point of view.

The relationships among category theory, set theory, various flavors of type theory, and other candidate foundations is not a simple matter to be summed up in a phrase.

I found an informative and insightful thread here.

Are category-theory and set-theory on the equal foundational footing?

The entire thread is well worth your time for anyone interested in contemporary foundations. In particular Derek Elkins's response is comprehensive and mind-expanding.

A few points. First, CT does not model ZFC. Rather, Elkins notes that "ETCS [Elementary theory of the category of sets] is equivalent to Bounded Zermelo set theory (BZ) which is weaker than ZFC."

Secondly, CT doesn't properly account for mathematical existence. This quote is a comment by Michael Greinecker:

"Set theory is full of axioms that guarantee that some things exist, which can be used to show that other things exist and finally that all the mathematical objects we want to exist do exist. Category theory doesn't really do that. You can formulate existence statements in categorical terms, but it is much less clear what kind of foundations category theory is meant to supply."

So it's not fair to say that you can "get set theory from category theory" or "do set theory within category theory." Those are facile statements. Facile means, "appearing neat and comprehensive only by ignoring the true complexities of an issue; superficial." That's apt.

On the other hand, can we do category within set theory? The conventional wisdom would be that classes are too big to be sets. The category of sets, for example is surely not a set. The category of Abelian groups is not a set. One way around this is to consider only "small" categories in which the objects and morphisms form sets. Another way is to say that we are agnostic as to whether a category contains "all" possible instances of an object type; but rather contains "enough" for any argument you need to make. I have seen this point of view expressed but don't have a reference at the moment.

Or we could just say that categories in general are proper classes, in the sense of "Predicate satisfiers that are too big to be sets." And then the argument is that since ZFC doesn't have proper classes, you can't do category theory within ZFC.

That is the argument. But what about a set theory like Morse-Kelly or Von Neuman-Bernays-Gödel set theory, two set theories that DO incorporate proper classes? Can you do category theory in those set theories? Good question.

This brings up a larger question: Which set theory, and which version of category theory? There are various flavors of each. The Stackexchange thread brings out this point in more detail. I truly hope people will read that page to get a sense of the many interrelated and nontrivial issues.

Set theory and category theory are not in a cage match to the death, as some seem to think. They're complementary ideas in a toolkit. It's like the programmers arguing over functional versus object-oriented. They're tools, not religions.

Thanks for the insights and the link.

Excellent reference! (Derek Elkins's response).

"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. — Mephist

Yes, I would say it is a trick of sophistry. To say of an attribute that it is "null", is to say that the attribute is non-existence. So to say that the attribute of extension is null, is to say that the attribute of extension is not there. So all you are saying is that the thing has the attribute of extension, but the attribute of extension is not there because it's null. Of course that's contradictory.

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. — Mephist

Do you not see, that if the supposed "two points" are really one point, then they are not two points at all, they are one point? That's directly from Leibniz' identity of indiscernibles. So if they are really one point, then there is no segment.

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". — Mephist

OK, we could take this route, but I think I've followed it before. Perhaps you could lead me to something new. Let's say that a set does not necessarily have extension, extension is not an essential feature. We'll say that a set may have extension or it may not have extension. Let's define "set" then. Isn't a set a collection of objects? Doesn't it appear contradictory to you, to speak of a collection of objects with no objects? If we define "set" as a possible collection of objects, such that the set is the defining terms rather than the actual objects, then no sets would actually have any objects and they would all be empty sets.

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. — Mephist

Wow, I find that a very strange way of looking at this. Instead of imagining a line exactly how it is defined, length with absolute purity, no width, you imagine a wide long thing, then subtract the width off it.

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! — Mephist

As I explained above, a set with no parts seems contradictory, and this is due to what a set is. Do you see the difference? A point is defined as having no parts. But a set has parts, according to what a set is, so the set with no parts doesn't make sense unless we changed what a set is. If a set is something other than a thing with parts, what is it?

: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: