None of the axioms of set theory mention 'set'. So objection to any axioms on the basis that they mention 'set' are ill-founded.

/

'set' is not a primitive of set theory. However it, may be defined in the language of set theory:

df: x = 0 <-> Ay ~yex

df: x is a class <-> (Ey yex v x=0))

df: x is set <-> (x is a class & Ey xey))

That definition also carries over to class theory such as Bernays class theory (more commonly known as 'NBG set theory').

But formally 'is a set' does not occur, rather it is an English phrase to stand for a formal predicate symbol. While 'set' is used to suggest our pre-formal notion of sets, it is not formally required. We could use 'zet' 'Zset' or myriad other words. Same for the phrase 'empty set'. So, while one may choose to argue that the notion of an empty set does not adhere to our usual understanding of what sets are, one should be careful not to charge that set theory is inconsistent on that basis.

The theorem and definition at issue are:

th: E!x Ay ~yex

df: Ax x = 0 <-> Ay ~yex.

No mention of 'set' or 'empty' or 'empty set'.

There is no need for an axiom regarding intersection. Intersection is just an instance of separation.

But political neutrality doesn't amount to reasonableness regarding which mathematics should be prioritised. — sime

Which math gets prioritized is of course a matter of historical contingency. Set theory in the 20th century, maybe category theory / type theory / topos theory / whatever in the 21st, and who knows what in the 22nd. I agree with you about this. However, you have expressed an antipathy to the axiom of choice that is not shared by category theorists. I'm not exactly sure where you are coming from. Are you a constructivist? They are gaining mindshare these days through the influence of computers. But Turing showed us that there are easily-stated problems that can not possibly be solved by a computer. So there will always be a place for nonconstructive math. In my "inappropriately trained" opinion, of course.

to the consternation of inappropriately trained mathematicians who resent not knowing constructive analysis. — sime

I got a chuckle out of this.

Like a mathematics department, nlab as an encyclopedia is obviously going to disseminate mathematics in a politically neutral fashion. Or perhaps i should have said, unlike a mathematics department — sime

Campus or departmental politics regarding subject matter and its relative "importance" can depend upon current interests of the faculty - and grant money. Sometimes a group of faculty convince administration to focus on or emphasize a particular topic those mathematicians are eager to pursue. Since math is a social activity directions may be decided by social interaction - and strong personalities. And, not least among the reasons for these choices, a department may decide to build around a well-known and accomplished colleague. The awarding of grant money is another story, similar to the above.

Category theory seems topical these days. My old clique - largely kerpunkt - hoped for a resurgence of interest in analytic continued fraction theory. The leaders have simply passed on. RIP

I'm a philosopher, — Metaphysician Undercover

A philosophy crank is more like it. You have zero familiarity with the 20th century literature on the philosophy of set theory. You haven't read Maddy, Quine, or Putnam. You have no interest in learning anything about the philosophy of set theory. When I mentioned to you the other day that Skolem was skeptical of set theory as a foundation for math, you expressed no curiosity and just ignored the remark. Why didn't you ask what his grounds were? After all he was one of the major set theorists of the early 20th century.

You don't seem to be able to engage in logical reasoning. You keep saying things that are demonstrably false, and when corrected, you simply repeat the same mistakes in your next post. You don't seem to understand elementary sentential logic, for example material implication and logical equivalence. About this more in a few paragraphs.

my game is to analyze and criticize the rules of other games. — Metaphysician Undercover

What do you think of the knight move in chess? When pressed, you tried to claim it's "physical" because chess sets are made of atoms. Childish sophistry.

This is a matter of interpretation. If you do not like that, then why are you participating in a philosophy forum? — Metaphysician Undercover

I'm here discussing the philosophy of math. I hope you don't think anything we've discussed is actual math. Do you think mathematicians sit around and talk about whether the empty set exists? And as I say, I know a little bit about the actual philosophical literature on the subject, and you haven't the slightest interest in it.

As much as you, as a mathematician are trying to teach me some rules of mathematics, I as a philosopher am trying to teach you some rules of interpretation. — Metaphysician Undercover

Is that why you can't work your way through a simple logical equivalence in sentential logic?

So the argument goes both ways, you are not progressing very well in developing your capacity for interpreting. — Metaphysician Undercover

Your cranky and ignorant ideas of math aren't subject to interpretation, only derision.

But if you do not like the game of interpretation, then just do something else — Metaphysician Undercover

Oh I'll be here to discuss the philosophy of math, if anyone is interested. Sadly you have nothing intelligible to say on the subject.

This is why the axiom of extensionality is not a good axiom. It states something about the thing referred to by "set", which is inconsistent with the mathematician's use of "set", as you've demonstrated to me. — Metaphysician Undercover

Nonsense. It's your own inability to follow an elementary exercise in logic that keeps you stuck.

We've already been through this problem, a multitude of times. That two things are equal does not mean that they are the same. — Metaphysician Undercover

That's not relevant here. However, two things that are mathematically equal are indeed the same.

That's why I concluded before, that it's not the axiom of extensionality which is so bad, but your interpretation of it is not very good. — Metaphysician Undercover

Mine is the perfectly standard interpretation, comprehensible to everyone who spends a little effort to understand it. Two sets are the same if and only if they have the same elements. Formally, if a thing is in one set if and only if it's in the other; which (as we will shortly see) includes the case where both sets are empty.

But I now see that the axiom of extensionality is itself bad. — Metaphysician Undercover

Like I say, you need to take that up with Ernst Zermelo. Or the authors of every set theory text in the world.

In case you haven't noticed, what I am interested in is the interpretation of symbols. — Metaphysician Undercover

How can that be? You are completely unable to understand even the most elementary symbolic reasoning.

And obviously the symbology of the axiom is not perfectly clear. If you can interpret "=" as either equal to, or the same as, then there is ambiguity. — Metaphysician Undercover

Are you saying that two things can be "the same" but not equal? Are you sure whatever you're on is legal in your jurisdiction?

Actually, I'm starting to see that this, what you claim in your vacuous argument, is not a product of the axiom of extensionality, but a product of your faulty interpretation. By the axiom of extensionality, a person on the moon is equal to a pink flying elephant, and you interpret this as "the same as". So the axiom is bad, in the first place, for the reasons I explained in the last post, and you make it even worse, with a bad interpretation. — Metaphysician Undercover

I'm going to walk you through this.

First, do you understand material implication? Material implication has the following truth table:


In other words "If 2 + 2 = 5 then I am the Pope" is a true material implication. Do you understand that? Do you agree? Do you have a disagreement perhaps?

Ok. Let $A$ be the set of pink flying elephants, and $B$ the set of people on the moon. For purposes of this exercise, let's assume these are not contingent. If you can't do that then make them the set of even numbers not divisible by 2, and the set of primes with nontrivial factorizations if you like.

Now I claim that for all $X$, it is the case that $X \in A \iff X \in B$. That is read as, "X is an element of A if and only if X is an element of B.

In sentential logic we break this down into two propositions: (1) If X is an element of A then X is an element of B; and (2) If X is an element of B then X is an element of A."

Now for (1). If X is a pink flying elephant, then it's a person on the moon. Is that true? Well yes. There are no pink flying elephants and there are no people on the moon. So this is line 4 of the truth table, the F/F case, which evaluates to True. So (1) is true.

How about (2)? Well the argument is exactly the same. If X is a person on the moon, then they are a pink flying elephant. As in (1), this is the False/False case of material implication.

Having shown both directions of the implication, we have that A and B are logically equivalent.

The axiom of extensionality says that if it's the case that X is in A if and only if X is in B, then A = B.

Therefore, by an exercise in elementary sentential logic, we see that the axiom of extensionality says that the two sets are the same.

Now here is what I know. I know that you are totally incapable of following this simple chain of logic. Or perhaps just unwilling. Either way, it's no longer my problem.

You really do not seem to be getting it. If, we can "use a predicate to form a set" as the axiom of specification allows, then it is not true that a set is characterized by its elements. — Metaphysician Undercover

Of course it is. We can use specification to prove the existence of some set; AND by extension if that set has the same elements as some other set, then the two sets are the same.

It's characterized by that predication. — Metaphysician Undercover

No no no. "Characterized" in this context is YOUR word but it's not what I've said and not what set theory says. Set theory says that (1) we can show some set exists using specification; and that set is also subject to extensionality: if some other set has the same elements, then the two sets are the same.

For example the set of "the first three positive integers is {1,2,3}. The set of the positive square roots of 1, 4, and 9 is (1, 2, 3}. Two distinct specifications. But the two sets have exactly the same elements; so they are the same set.

Specification lets you show some set exists. Extensionality tells you when some set is equal to some other set.

I can not for the life of me figure out why you won't get this. I will say that I'm not sure how we got onto this particular subtopic, and that I'm finding it tedious in the extreme, so I probably won't be replying back much as long as you continue to be trollishly repeating this fallacious line of argument.

The two are mutually exclusive, inconsistent and incompatible. — Metaphysician Undercover

No, they are quite independent of each other. Specification is one way (out of several others) to show that a given set exists. Extensionality tells you when that set is equal to some other set.

Specification allows for a nonempty set, I have no problem with this. But to say that this set is characterized by its elements is blatantly false. It has no elements, and it is characterized as having zero elements, an empty set. So it's not characterized by its elements, it's characterized by the number of elements which it has, none. . — Metaphysician Undercover

I just gave you a formal proof to the contrary. An object is in A if and only if it's in B. If A and B happen to be empty, that is a true statement. Therefore by extensionality, A = B. Or in words, "There's only one empty set."

Yes, this is the problem with the axiom of extension, in its portrayal of the empty set. It is saying that if two specified sets each have zero elements, then "the elements themselves" are equal. — Metaphysician Undercover

You're confusing yourself because you are unable/unwilling to do a little basic sentential logic, or perhaps that you do not understand material implication.

Which is it?

However, there are no such elements to allow one to judge the equality of them. — Metaphysician Undercover

You are confusing yourself because you can't/won't follow the symbolic logic. For all X, X is in A if and only if x is in B. That's as clear as can be.

Tell me, do you disagree or happen to not know basic sentential logic? Do you understand that "if 2 + 2 = 5 then I am the Pope" is true?

So there is no judgement — Metaphysician Undercover

No middle e in judgment. Please make a note of it.

that "the elements themselves" are equal, because there are no elements to judge, and so the judgement of cardinal equivalence, that they have the same number of elements, zero, is presented as a judgement of the elements themselves. — Metaphysician Undercover

X is in A if and only if X is in B. Can it be the case that you don't actually know basic sentential logic, and don't understand material implication, and therefore cannot understand the content of the axiom of extensionality? That would explain a lot.

You ought to recognize, that to present a judgement of cardinal equivalence, as a judgement of the elements themselves, is an act of misrepresentation, which is an act of deception. I know that you have no concern for truth or falsity in mathematical axioms, but you really ought to have concern for the presence of deception in axioms. — Metaphysician Undercover

You interpret your own ignorance as deception by others. Pretty funny.

Now, do you agree, that when there are no elements, it makes no sense to say that the elements themselves are respectively equal? — Metaphysician Undercover

They are vacuously respectively equal. I see that it must be the case that you do not actually understand material implication. That explains quite a lot.

The axiom of extensionality, X is in A if and only if X is in B, is the same as "If 2 + 2 = 5 then I am the Pope; AND if I am the Pope then 2 + 2 = 5." I see that you truly don't get this. @Meta my friend you need to go study up on basic sentential logic. As a self-proclaimed philosopher you are missing the very basics.

What is really being judged as equal is the cardinality. They both have zero elements. — Metaphysician Undercover

I see why you think this. It's because you don't understand material implication and logical equivalence.

No, the axiom of extensionality does not tell us when two sets are the same, that's the faulty interpretation I've pointed out to you numerous times already, and you just cannot learn. It tells us when two sets are equal. — Metaphysician Undercover

Someone a while back pointed out that in math, equality is extensional. You are taking it intentionally. But whatever. It's not relevant to the fact that we've discovered that you can't do basic sentential logic. That's something you should remedy at once.

That faulty interpretation is what enables the deception. Equality always indicates a judgement of predication, and in mathematics it's a judgement of equal quantity, which you call cardinal equivalence. When you replace the determination of the cardinality of two empty sets, "equal", with "the same", you transfer a predication of the set, its cardinality, to make a predication of its elements, "the same as each other". I believe that's known as a fallacy of division. — Metaphysician Undercover

Throwing more crap on the wall doesn't remedy your inability to understand a simple logical argument.

Well, "pink flying elephants" was your example, and it's equally contingent. — Metaphysician Undercover

For purposes of this discussion, we take the two predicates as absolute and not contingent. You're just raising this red herring to sow confusion. The only one confused here is you.

The issue of temporally contingent propositions raises a completely different problem. The only truly necessary empty set is the one specified as "the empty set". As your examples of square circles and married bachelors show, definitions and conceptual structures change over time, so your assertion that mathematics has no temporally contingent propositions is completely untrue. It may be the case that "the empty set" will always refer to the empty set, necessarily, but how we interpret "empty" and "set" is temporally contingent. So temporal contingency cannot be removed from mathematics as you claim. This is the problem of Platonic realism, the idea that mathematics consists of eternal, unchanging truths, when in reality the relations between symbols and meaning evolves. — Metaphysician Undercover

You can throw all the crap on the wall you like. I'm done cleaning it up. Change the subject, I'm done with this. Go read a book on logic and then work through the axiom of extensionality, which frankly is quite simple.

A philosophy crank is more like it. You have zero familiarity with the 20th century literature on the philosophy of set theory. You haven't read Maddy, Quine, or Putnam. You have no interest in learning anything about the philosophy of set theory. When I mentioned to you the other day that Skolem was skeptical of set theory as a foundation for math, you expressed no curiosity and just ignored the remark. Why didn't you ask what his grounds were? After all he was one of the major set theorists of the early 20th century. — fishfry

If some of this is relevant to the points I've made, then provide some quotes or references. Otherwise what's the point in mentioning something which is not relevant?

Mine is the perfectly standard interpretation, comprehensible to everyone who spends a little effort to understand it. Two sets are the same if and only if they have the same elements. Formally, if a thing is in one set if and only if it's in the other; which (as we will shortly see) includes the case where both sets are empty. — fishfry

You haven't addressed the point. To have "the same elements" requires a judgement of elements. Having no elements is not an instance of having elements, and there are no elements to be judged. An empty set has no elements therefore two empty sets do not have the same elements, because they both have no elements. Therefore two empty sets are not the same.

Are you saying that two things can be "the same" but not equal? Are you sure whatever you're on is legal in your jurisdiction? — fishfry

We've been through the law of identity before, and you still show no desire to understand it.. No two things are the same, according to that law. If it's "the same", then there is only one thing. That's what "the same" refers to according to the law of identity, one and the same thing. The law of identity dictates that we use "same" to refer to only one thing, so it is impossible that two distinct things are the same. However, two distinct things may be equal. Therefore "equal" is not synonymous with "the same".

In other words "If 2 + 2 = 5 then I am the Pope" is a true material implication. Do you understand that? Do you agree? Do you have a disagreement perhaps? — fishfry

Sorry fishfry, but you'll need to do a better job explaining than this. Your truth table does not show me how you draw this conclusion.

Now I claim that for all XX, it is the case that X∈A⟺X∈BX∈A⟺X∈B. That is read as, "X is an element of A if and only if X is an element of B.

In sentential logic we break this down into two propositions: (1) If X is an element of A then X is an element of B; and (2) If X is an element of B then X is an element of A."

Now for (1). If X is a pink flying elephant, then it's a person on the moon. Is that true? Well yes. There are no pink flying elephants and there are no people on the moon. So this is line 4 of the truth table, the F/F case, which evaluates to True. So (1) is true. — fishfry

You have already determined that there are no elements in both sets A and B. This is predetermined, they are empty sets. So your starting point, "(1) If X is an element of A then X is an element of B" is not relevant, there are not elements. That's like saying if C and D are both green, when you've already determined that they are not green. It's an irrelevant premise, and your entire appeal to material implication is unacceptable. We already know that there are no elements of both A and B, so that premise concerning the elements of A is not applicable. The two premises "A is an empty set", and "if X is an element of A" are fundamentally contradictory.

I'll give you credit for at least addressing the point now. It was a nice try, but your attempt is a failure.

I just gave you a formal proof to the contrary. An object is in A if and only if it's in B. If A and B happen to be empty, that is a true statement. — fishfry

As explained above, it's actually an irrelevant, and inapplicable statement. And it can only be applied under contradiction. When a set has been determined as empty, then to talk about objects within that set is contradiction. So an attempt to apply this statement to empty sets is contradictory. Look at what you're saying 1)There are not any objects in set A. 2) An object is in A if... See the contradiction? When you've already designated A as having no objects, how does it make sense to you to start talking about the condition under which there is an object in A? Do you agree that there is contradiction here?

As you can see, your attempt at a formal proof is a failure due to contradicting premises.

You interpret your own ignorance as deception by others. Pretty funny. — fishfry

The thing with this type of deception, is that you can either recognize it as deception, and reject it, or you can join it, and become one of the deceivers. This is why mathematics is similar to religion (Tones will disagree), the authors have good intentions, but once falsity is allowed into the premises, deception is required to maintain respect for the premises amongst the masses. When the deception has been pointed out to you, as I have, then you can either reject it and work toward dismantling the system which propagates it, or you can support it with further deception. You it appears, are choosing to be one of the deceivers.

For purposes of this discussion, we take the two predicates as absolute and not contingent. You're just raising this red herring to sow confusion. The only one confused here is you. — fishfry

This thoroughly supports my argument. If the empty sets are necessarily empty, absolute and not contingent, then to talk about the conditions under which there are elements in those sets is very clearly contradiction.

A contradiction is the conjunction of a statement and its negation.

A contradiction is the conjunction of a statement and its negation. — TonesInDeepFreeze

Like for example, when someone says "an empty set has no elements", and also says "the elements of the empty set A are the same elements as the elements of the empty set B". The former says the empty set has no elements, the latter states its negation "the elements of the empty set...".

Or, even if one were to say that an empty set has no elements, absolute and not contingent, and then states the conditional "if X is an element of the empty set...", as if the set is only contingently empty, that would also constitute a statement and its negation.

Don't you agree?

A conjunction of a statement and its negation is of the form

P & ~P

where P is any statement.

A conjunction of a statement and its negation in the language of set theory is of the form

P & ~P

where P is a formula in the language of set theory.

Set theory is inconsistent if and only if there is a such a conjunction that is a theorem of set theory.

The thing with this type of deception — Metaphysician Undercover

I'm going to let you have the last word. I'm out. But for the record, can you please name the specific individuals involved in this deception? We need their names to hold them accountable at the Stalinist show trials to begin soon. Cantor? Zermelo? Mrs. Zermelo, who was pro choice? Abraham Fraenkel? Should John von Neumann be included? He did invent mathematical economics and worked on the hydrogen bomb, but ... he DID do foundational work in set theory as well. Might was well include him on the list. How about the modern set theorists Solovay, Magidor, Shelah? Or the contemporary ones like Woodin and Hamkins? The modern philosophers of set theory like Quine, Putnam, and especially Maddy? Are they all involved in this deception? Please be specific, we need to know how many cells to reserve at Gitmo.

In other words "If 2 + 2 = 5 then I am the Pope" is a true material implication. Do you understand that? Do you agree? Do you have a disagreement perhaps?
— fishfry

Sorry fishfry, but you'll need to do a better job explaining than this. Your truth table does not show me how you draw this conclusion. — Metaphysician Undercover

You don't know material implication? You will find "my" truth table on that page. How deep exactly is your ignorance? "My" truth table? Do you really mean to say you never saw this before? I guess I don't understand how that could be. Honestly, in all your time on this forum and presumably studying the philosophy of math, you never saw basic sentential (aka propositional) logic?

Here you go. https://en.wikipedia.org/wiki/Propositional_calculus

You can have the last word. Though after calling modern mathematics a "deception" and admitting that you are unfamiliar with the truth table for material implication, I don't see how you could top what's gone before.

Thanks for the chat. All the best.

I believe in our universe an infinite ocean would inescapably collapse into a black hole