The question is exact.

What is supposed to be THE order of the set whose members are all and only the famous four bandmates in the Beatles?

And woah! The poster is now saying the exact opposite of what he said for dozens of posts in this forum.

Over and over, the poster argued against the axiom of extensionality on the grounds that there is THE ordering of a set. Yes, I do remember.

And now he's denying he said that sets do have a certain ordering that is the ordering of the set.

The poster is flat out lying.

And completely reversing his previous argument. He says now that sets do not have an inherent order so that the law of identity does not apply to them. But he had been saying that sets do have an inherent order so that that order must be regarded when we evaluate self-identity. But it is the axiom of extensionality, which the poster denies, that provides that sets do not have an inherent order. And the poster denied the axiom of extensionality on the basis that sets do have an inherent order.

Then the poster mentions "things", though he denies that mathematical abstractions are things. But now he talks about sets as things. And he says their identity depends on their ordering but that a set can't be identical with itself. (What? A set can't be identical with itself?) Why can't a set be identical with itself? Now he says it's because a set does not have a certain ordering. Not only does it make no sense that a set is not identical with itself, it makes no sense that the reason a set is not identical with itself is that it does not have a certain ordering, especially when the poster's argument was that the reason sets fail the law of identity is that they do have a certain ordering.

The poster is not only lying, but he's very confused.

For any set that has more than one thing as a member (whether mathematical things or material things), there is more than one ordering of that set. That does not contradict the law of identity.

So put in terms the poster just used:

The set whose members are all and only the famous four bandmates in the Beatles.

That set is identical with itself (the law of identity applied to that set). But there is more than one ordering of that set. If a set being identical with itself required having one particular ordering, then no set with more than one member could be identical with itself. But the set that whose members are the bandmates in the Beatles is identical with itself.

So again the question for the poster: What is supposed to be THE one particular ordering of the Beatles that permits the set whose members are the bandmates in the Beatles to be identical with itself? Or, it seems that the poster's answer now is that sets are not identical with themselves. But that itself violates the law of identity, since the law of identity is that every thing is identical with itself - and whether material things or mathematical things.

Here's how it looks abstracted to a dialogue:

MU (many times a while back): The axiom of extensionality is wrong because sets have a certain order but the axiom allows that sets are equal without regard to their order.

TIDF (then and now): Sets of more than one member have more than one order. If every set has just one order, then, for example, what is the order of the set whose members are all and only the famous bandmates in the Beatles?

MU (many times): [silence]

TIDF (many times): Still interested in an answer.

MU (now): Sets do not have a certain order. And that is why sets are not identical with themselves.

TIDF (now): Many times you said that sets do have a certain order so that the axiom of extensionality is wrong. Now you say that sets do not have a certain order and that that is why they are not identical with themselves. But it's a violation of the law of identity that there are things that are not identical with themselves. So, sets, whatever they are, are identical with themselves. And from identity theory with the axiom of extensionality we have that sets are identical if and only if they have the same members, which denies that sets have only one certain order, and the law of identity is upheld not contravened. You are dishonest, self-contradictory and confused, and you don't know anything about this subject.

And now he's denying he said that sets do have a certain ordering that is the ordering of the set. — TonesInDeepFreeze

Over and over, the poster argued against the axiom of extensionality on the grounds that there is THE ordering of a set. Yes, I do remember.

And now he's denying he said that sets do have a certain ordering that is the ordering of the set. — TonesInDeepFreeze

Tones, I argued that the axiom of extensionality does not indicate identity in a way which is consistent with the law of identity, because the identity of a thing (by the law of identity) includes the order of the constituent elements, while the identity of a set (by the axiom of extensionality) does not include the order of the elements. Therefore, i conclude that "identity" in set theory, as indicated by the axiom of extensionality is inconsistent with the law of identity. That sort of "identity", found in set theory, is a violation of the law of identity. If you really think that I was arguing the opposite to this, then I'm sure you can provide a reference.

If anyone is lying, it is you in your misrepresentation of what I argued in the past. However, I do not think you are lying, I think you have difficulty understanding English. Perhaps you have been overworking yourself, being immersed in mathematical symbols for so long that you no longer understand common language.

The poster is very mixed up and adding lies to the ones he's already committed.

This is revisionist: [strikethrough in edit, since 'revisionist' is not the right word there.]

"the identity of a thing (by the law of identity) includes the order of the constituent elements, while the identity of a set (by the axiom of extensionality) does not include the order of the elements."

Yes, by the axiom of extensionality, sets do not have an inherent order. But in the past the poster argued that therefore the axiom of extensionality is wrong, because there IS the ordering of a set. That is why I asked: What is THE ordering of the set whose members are the bandmates in the Beatles.

Moreover:

A thing that has elements (or at least is made up only of its elements) is a set. So if things have an ordering that is "THE" [emphasis added] order of its elements, then, AGAIN, that is to say that, for sets, there is THE ordering of a set.

As I said:

Originally, his longstanding claim was that there is "THE" [emphasis added] ordering of a set . And that was the basis for him rejecting the axiom of extensionality. So I challenged by asking what does he claim is THE ordering of the set whose members are all and only bandmates in the Beatles (keeping in mind that there are 24 different orderings of that set, as, in general any set with n number of members has n! orderings).

But yesterday he said that sets do "NOT" [emphasis added] have an inherent ordering. So that was a reversal from his previous longstanding claim. But, even worse, yesterday he contradicted himself on the matter of whether there is "THE" ordering of a set.

Today he's back to:

There IS "THE" ordering of "things [with] constituent elements".

And that is what I said that he had said originally.

/

And contrary to the poster with his wildly ersatz, ignorant, stubborn, and mixed up ideas not just of mathematics but of everyday notions, I do recognize the sense of sets in everyday discourse. For example:

There is the set whose members are the bandmates in the Beatles. Suppose a painter asks for the Beatles to sit for a portrait - left to right in their order. Then they would say, "WHAT order?" There is no THE order of the Beatles. There are 24 of them.

Even if a person doesn't know to calculate that there are twenty-four orderings of a set with four members, a person does understand that there is not just one ordering of the set.

If a schoolmaster says, "Now, children, line up in your order." Even children would have the sense to say, "What order? Order by height? Order by age? Order by grade point average? Or what?".

But the poster still cannot grasp this fact of common understanding, though he presumes to take refuge in a sense of "common" language.

Sheesh!

But in the past the poster argued that therefore the axiom of extensionality is wrong, because there IS the ordering of a set. — TonesInDeepFreeze

Blah, blah, blah., so much hot air. Show us your evidence. Fishfry showed me years that there is no necessary order to the elements of a set. That's definitional, why would I argue against it?

What I argue, as I've argued for years, is that the so-called "identity" of set theory, is inconsistent with, therefore in violation of, the law of identity. And, the fact that there is no order to the elements of a set is very good evidence for what I argue. However, I do not argue that set theory is "wrong" on account of this violation, because some philosophers suppose the law of identity to be unacceptable. I argue that people like you, who insist that identity in set theory (or what you call identity theory), is consistent with the law of identity, are wrong.

So fly away now, Mr. Balloon, because you expulsions of hot air are threatening to blow the thread off track. .

It's not hot air that the poster's claim was (and is now back) that there is THE ordering of a set.

Evidence includes the poster's penultimate post: "I argued that the axiom of extensionality does not indicate identity in a way which is consistent with the law of identity, because the identity of a thing (by the law of identity) includes the order of the constituent elements, while the identity of a set (by the axiom of extensionality) does not include the order of the elements."

And revisionist: [strikethrough in edit, since 'revisionist' is not the right word there.]

"I do not argue that set theory is "wrong" on account of this violation, because some philosophers suppose the law of identity to be unacceptable."

It is not at all typical that philosophers of mathematics who are interested in set theory suppose the law of identity to be unacceptable.

What specific philosophers is the poster referring to?

And what the poster says makes no sense, again. He endorses the law of identity (no?). And even one post ago he falsely claimed that set theory is not consistent with the law of identity. But now, revisionistically, he says he doesn't fault set theory, and that the reason he doesn't fault set theory is that some philosophers deny the law of identity. But if he endorses the law of identity, then it would make no sense for him to let set theory off the hook on the basis that there are philosophers who disagree with his endorsement of the law of identity. The poster is brazenly illogical, again.

/

The law of identity is that a thing is identical with itself. It is an axiom of identity theory, which is presupposed by set theory. Not only is set theory consistent with the law of identity, but the law of identity is one of the pre-axioms of set theory.

/

And the poster is back to claiming that a set has an ordering that is THE ordering of the set. And still he does not address the natural rejoinder: If a set has an ordering that is THE ordering of the set, then which of the 24 orderings of the set whose members are the bandmates in the Beatles is THE ordering of that set?

/

As to the track here, the poster's position regarding mathematics in this context is fairly paraphrased as:

1/2 + 1/2 is not 1, because you can't cut pie without some crumbs falling around so that the resulting two pieces of pie are not precisely the same size.

No exaggeration. The poster's notion is that that 1/2 + 1/2 is not 1, since the '1/2 + 1/2' is not '1' (though the particular example was different). (And even the most utterly ridiculous irrelevancy that with '1 = 1', the first occurrence of '1' is not the second occurrence of '1'.)

And now, moreover that the identity assertion fails, because, for example, pie cutting is not exact.

What specific philosophers is the poster referring to? — TonesInDeepFreeze

Hegel for example:
https://thephilosophyforum.com/discussion/9078/hegel-versus-aristotle-and-the-law-of-identity/p1

If Hegel rejects the law of identity, but the poster endorses it, then it makes no sense for the poster to invoke Hegel as vindicating violations of the law of identity thus excusing the set theory that the poster abhors.

Moreover, the context is the law of identity vis-a-vis mathematics. As I said, it is not at all typical that philosophers of mathematics who are interested in set theory suppose the law of identity to be unacceptable. So it is not typical for mathematicians and philosophers of mathematics to vindicate set theory on the basis of denying the law of identity. Quite the contrary, it is typical for mathematicians and philosophers of mathematics to accept or endorse the law of identity, especially as the law of identity is one of the axioms used in set theory.

Moreover, the context is the law of identity vis-a-vis mathematics. — TonesInDeepFreeze

The law of identity in its historical form is ontological, not mathematical. Mathematics might have its own "law of identity", based in what you call "identity theory", but it's clearly inconsistent with the historical law of identity derived from Aristotle. He proposed this principle as a means of refuting the arguments of sophists such as those from of Elea, (of which Zeno was one), who could use logic to produce absurd conclusions.

Discussion with you about this is pointless because you make statements like the one above, where you acknowledge the difference between the mathematical concept of "identity" and the ontological concept of "identity", but you claim that the only relevant concept of "identity" is the mathematical one.

Of course, relevance depends on one's goals, and truth is clearly not one of yours.

The poster wrote:

"you claim that the only relevant concept of "identity" is the mathematical one"

That is false. I've said very much the opposite in this forum. Of course identity is treated in philosophy aside from mathematics and as an everyday notion. And especially in philosophy and in certain alternative mathematics there are a great many differing views of the subject, all of which are we may benefit by study and comparison.

I wrote:

"Moreover, the context is the law of identity vis-a-vis mathematics."

That was in response to the poster's own claims about the law of identity vis-a-vis mathematics. That is, the context of this part of the discussion with the poster has been his attack that mathematics is incompatible with the law of identity. I have never at all claimed that mathematics has sole authority regard the subject of identity. Rather, I have shown (in posts in this forum) how the poster's attacks on mathematics vis-a-vis identity are ill-founded.

The poster disputes that mathematics upholds that law of identity. But the law of identity, both symbolically and as understood informally, is an axiom of classical mathematics. The poster has two prongs in reply:

One prong in the poster's reply is (as best I can summarize): Mathematicians may claim to state the law of identity, but those statements are incompatible with the actual law of identity, since mathematics regards numbers and such* as objects but those are not objects, and the axiom of identity pertains only to objects.

* If I say 'numbers and other things' or 'mathematical things', then that is tantamount to referring to objects since 'object' and 'thing' are synonyms. So, if we may not refer to mathematical objects then we should not even use the word 'thing' regarding, well, mathematical things.

The other prong in the poster's reply is: Mathematics regards sets as identical if and only they have the same members, but that ignores the orderings of the members of the sets, and the ordering of a set is crucial to the identity of a set, so sets are not identical merely on account of having the same members.

(1) OBJECTS.

If I'm not mistaken, objection to referring to numbers and such as objects is something like this: There are only physical, material or concrete objects, but mathematics regards numbers and such as abstractions.

But, without prejudice as to whether numbers and things are not physical, material or concrete, in mathematics, philosophy and in everyday life, people do refer to numbers and such as objects. It's built into the way we speak, as numbers and such are referred to by nouns and are the subject in sentences. If we weren't allowed to speak of numbers and such as objects then discourse about them would be unduly unwieldy.

But one might counter, "People talk like that, or think they need to talk like that, but that doesn't entail that it is correct that they do." To that we might say, "Fair enough. So when we say 'mathematical object' we may be regarded, at the very least, as using the word 'object' as "place holder" in sentences where it would be unduly unwieldy otherwise. The mathematical formulations themselves do not use the word 'object'. One could study formal mathematics for a lifetime without invoking the word 'object'. But to communicate informally about mathematics, it would be unwieldy to not be allowed to use the word 'object'. Moreover, when 'object' is used in that "place holder" way, one may stipulate that one does so without prejudice as to whether mathematical objects are to be regarded as more than abstractions, concepts, ideals, fictions, hypothetical "as if" things, platonic things, values of a variable, members of a domain of discourse, etc.

And it seems to me that the notion of 'object' itself may be regarded as primitive - basic itself to thought and communication. Any explication of the notion of 'object' would seem fated to eventually relying on the notion of 'object'.

In regards all of the above: Philosophy of mathematics enriches understanding and appreciation of mathematics, but one can study mathematics for a lifetime without committing to any particular philosophy about it. Moreover, one can study philosophy for a lifetime without committing to any particular philosophy. One may critically appreciate different philosophies without having to declare allegiance to a certain one. And one may use the word 'object' in a most general sense, even in a "place holder" role, without saying more about then that we may regard one's usage without prejudice as to how it should or should not be explicated beyond saying, "whatever sense of object that you may have about the "things" mathematics talks about".

The poster wrote:

"The law of identity in its historical form is ontological, not mathematical. Mathematics might have its own "law of identity", based in what you call "identity theory", but it's clearly inconsistent with the historical law of identity derived from Aristotle."

(2) LAW OF IDENTITY

To start, from the above quote, should the poster be charged here with argumentum ad antiquitatem? Even if not, there are more things to say.

Just to note, if I'm not mistaken, Aristotle's main comments about identity are in 'Metaphysics'. I don't have an opinion whether that's properly considered ontology.

The law of identity is usually stated as "A thing is identical with itself", or "A thing is itself" or similar.

Through history identity became an important subject in logic, philosophy and mathematics.

In logic, two central ideas emerged: The law of identity and Leibniz's identity of indiscernable and indiscernibility of identicals.

Eventually, mathematical logic provided a formal first order identity theory:

Axiom. The law of identity.

Axiom schema. The indiscernibility of identicals.

(The identity of indiscernbiles cannot be formulated in a first order language if there are infinitely many predicates, but it can be formulated in a first order language if there are only finitely many predicates.)

Along with the axioms, a semantics is given that requires that '=' stand for 'is identical with'. That is taken as 'is equal to', 'equals', 'is', 'is the same as' or any cognate of those.

So when we write:

x = y

we mean:

x is identical with y

x is equal to y

x equals y

x is y

x is the same as y

However, the poster, in all his crank glory, continues to not understand:

x = y

does NOT mean:

'x' is identical with 'y'

'x' is equal to 'y'

'x' equals 'y'

'x' is 'y'

'x' is the same as 'y'

but it DOES mean:

what 'x' stands for is identical with what 'y' stands for

what 'x' stands for is equal to what 'y' stands for

what 'x' stands for equals what 'y' stands for

what 'x' stands for is what 'y' stands for

what 'x' stands for is the same as what 'y' stands for


The law of identity is:

For all x, x = x

And by that we mean:

For all x, x is identical with x


And that does not depend on what kind of objects x ranges over. WHATEVER you regard a term 't of mathematics to refer to, the referent of 't' is identical with itself.

Classical mathematics does uphold the law of identity as it has been ordinarily understood in philosophy and as it came through Aristotle.

(3) EXTENSIONALITY

Still, the poster cannot say what THE ordering is of the set whose members are the bandmates in the Beatles. So, still, his claim (every set has and order that is THE order of the set) is not sustained, thus still unsustained is his second prong mentioned above.

The law of identity in its historical form is ontological, not mathematical. Mathematics might have its own "law of identity", based in what you call "identity theory", but it's clearly inconsistent with the historical law of identity derived from Aristotle. He proposed this principle as a means of refuting the arguments of sophists such as those from of Elea, (of which Zeno was one), who could use logic to produce absurd conclusions. — Metaphysician Undercover

I can't believe this thread is still going. I see you've hijacked it to your hobby horse.

For my edification, can you explain the above paragraph?

My understanding of your point is this, and do correct me if I'm wrong.

My concept of your thesis: The law of identity says that a thing is equal to itself. Mathematical equality is not metaphysical identity. Therefore math is wrong. Or something.

But nobody claims mathematical equality is identity. It may be spoken of that way in casual conversation, and by mathematicians who have not given the matter any thought and mean nothing by it.

When pressed, a mathematician would readily admit that mathematical equality is nothing more than a formal symbol defined within ZF set theory in the logical system of first order predicate logic. It's not actually the same as mathematical identity. It's not the same as anything.

If we like, we can visualize that equality is identity. Why not? Everything in our mathematical world is a set; if there are things that are identical but not equal as sets, they're entirely out our consciousness. So they often think of it that way. But they don't mean it as any kind of metaphysical thesis. It's just a manner of speaking, like jargon in any field.

Any logically or philosophically-oriented mathematician will immediately concede the point; and the rest would have no opinion at all. Most working mathematicians don't spend any times thinking about whether mathematical equality is logical identity. The question doesn't enter their minds.

So that's what I understand about your thesis, and I don't know anything about the Eleatics. What is your point with all this?

Discussion with you about this is pointless because you make statements like the one above, where you acknowledge the difference between the mathematical concept of "identity" and the ontological concept of "identity", but you claim that the only relevant concept of "identity" is the mathematical one. — Metaphysician Undercover

I'm just jumping into this convo between you and @TonesInDeepFreeze, and I can't speak for why he wrote what he did.

But I can tell you that I would agree with what he said, in the context of mathematics.

That is, if I'm a mathematician, all of the objects I deal with in my life are sets of one kind or another, and we know what equality for sets, we defined it via the axiom of extension.

You would be fun in set theory class. You're entirely hung up on the very first axiom. "Class, Axiom 1 is the axiom of extensionality. It tells us when two sets are equal." You, three years later: "But that's not metaphysical identity! You mathematicians are bad people. And you don't understand anything!" And your professor goes, Meta, We still have a countable infinitely of axioms to get through! Can we please move on?

But axioms don'g mean anything. They're just rules in a formal game, like chess. As I say, if you asked a mathematician if mathematical equality is metaphysical identity, a few of them would have an educated opinion about the matter and they'd immediately agree with you. The rest, the vast majority, wouldn't understand the question and would be annoyed that you interrupted them.

Of course, relevance depends on one's goals, and truth is clearly not one of yours. — Metaphysician Undercover

Such a civil one. It would be impolite even if you weren't also completely wrong.

In practise the math always refers to something. — Metaphysician Undercover

I found this in an old post of yours. It it exactly your misunderstanding.

Nobody claims that math refers. That's your straw man.

Now math can be useful. So we often play the game of (1) Assume math refers to this particular situation; and (2) Use math to improve your control of the situation, whatever it is.

Just because we constantly apply math as if it refers to the world; and just because it so often turns out to be really useful; does not mean that math itself refers to the world. That's the brilliant essence of math. Math refers to nothing; but is locally useful in almost everything.

You just don't get that. You're fighting a straw man of your own creation.

Clearly "identity" by the law of identity includes the order of a thing's elements, as it includes all aspect of the thing, even the unknown aspects. So the ordering of the thing's elements is therefore included in the thing's identity, unlike the supposed (fake) "identity" stated by the axiom of extensionality. — Metaphysician Undercover

I remember fondly when I spent weeks trying to explain order theory to you, back when I thought you were trying to understand anything. You are still at this. If two sets have the same elements and the same order, they are equal as ordered sets. It's just about layers of abstraction, separating out concepts. First you have things, then you place them in order.

Somehow this offends you. Why?

I see you've hijacked it to your hobby horse. — fishfry

That was said to Metaphysician Undercover.

Actually, I am the one who took up his misconception that sets have an inherent order. I don't consider that "hijacking", since his posts in this thread need to be taken in context of his basic confusions about mathematics, as mathematics has been discussed here.

nobody claims mathematical equality is identity — fishfry

In ordinary mathematics, '=' does stand for identity. It stands for the identity relation on the universe.

For terms 't' and 's', 't = s' is true if and only if what 't' stands for is identical with what 's' stands for.

When pressed, a mathematician would readily admit that mathematical equality is nothing more than a formal symbol defined within ZF set theory in the logical system of first order predicate logic — fishfry

An extreme formalist would say that. There is no evidence I know of that more than a very few mathematicians take such an extreme formalist view. Indeed, mathematicians and philosophers of mathematics often convey that they regard mathematics as not just formulas. Not even Hilbert, contrary to a false meme about him, said that mathematics is just a game of symbols.

And the semantics of first order logic with identity usually do require that '=' stands for the identity relation.

That was said to Metaphysician Undercover. — TonesInDeepFreeze

Yes I understood that. I haven't been in this thread in a couple of weeks and when I checked it out, @Metaphysician Undercover had evidently introduced his favorite theme, that mathematical equality is not metaphysical identity, which seemed a little afar from the original topic.

Actually, I am the one who took up his misconception that sets have an inherent order. — TonesInDeepFreeze

Yes I did actually understand that! I was just startled that @Meta was still going on about order being an inseparable and inherent aspect of a set, when I had already had such a detailed conversation with him on this subject several years ago. I did actually realize you were quoting him -- I was just surprised to see him still hung up on that topic.

I don't consider that "hijacking", since his posts in this thread about tasks need to be taken in context of his basic confusions about mathematics, as mathematics has been discussed here. — TonesInDeepFreeze

@Meta's basic confusions in math are too big to to be placed within context. He's completely unwilling to meet any mathematical idea on its own terms.

What? In ordinary mathematics, '=' does stand for identity. It stands for the identity relation on the universe. — TonesInDeepFreeze

Ah ... you said that? Well I understand why my friend Meta is unhappy with you then. I am not sure if I agree with your statement.

After all if = is the identity relation on the universe, why does ZF need to redefine it then? Is the = of ZF the same as the identity relation on the universe?

I do not think so. Because the ZF version is a definition. It's a defined symbol that wasn't defined before.

Answer me this, maybe I'll learn something. Suppose X and Y are objects in the universe, but they are not sets?

Well in ZF they don't exist. And even if they did, how would you define X = Y?

After all if = is the identity relation on the universe, why does ZF need to redefine it then? — fishfry

There are three ways we could approach for set theory:

(1) Take '=' from identity theory, with the axioms of identity theory, and add the axiom of extensionality. In that case, '=' is still undefined but we happen to have an additional axiom about it. The axiom of extensionality is not a definition there. And, with the usual semantics, '=' stands for the identity relation. It seems to me that this is the most common approach.

(2) Don't take '=' from identity theory.

Definition: x = y <-> Az(z e x <-> z e y)

Axiom: x = y -> Az(x e z -> y e z)

(3) Don't take '=' from identity theory.

Definition: x = y <-> Az(z e x <-> y e z)

Axiom: Az(z e x <-> z e y) -> Az(x e z -> y e z)

With (2) and (3), yes, '=' could stand for an equivalence relation on the universe that is not the identity relation. But it seems to me that even in this case, we'd stipulate a semantics that requires that '=' stands for the identity relation. And I think it's safe to say that usually mathematicians still regard '1+1 = 2' to mean that '1+1' stands for the same number that '2' stands for, and not merely that they stand for members in some equivalence relation, and especially not that it's just all uninterpreted symbols.

Suppose X and Y are objects in the universe, but they are not sets? — fishfry

Depends on what you mean by 'set' and what meta-theory is doing the models.

In set theory, contrary to a popular notion, we can define 'set':

x is a class <-> (x =0 or Ez z e x)

x is a set <-> (x is a class & Ez x e z)

x is a proper class <-> (x is a class & x is not a set)

x is an urelement <-> x is not a class

Then in ordinary set theory we have these theorems:

Ax x is a class

Ax x is set

Ax x is not a proper class

Ax x is not an urelement

If our meta-theory for doing models has only sets, then all members of universes are sets.

If our meta-theory for doing models has proper classes, still a universe is a set (proof is easy by the definition of 'model'). And no proper class is a member of a set.

If our meta-theory for doing models has urelements, and '=' stands for the identity relation, then the axiom of extensionality is false in any model that has two or more urelements in the universe or has the empty set and one or more urelements in the universe.

@Metaphysician Undercover, @TonesInDeepFreeze, Ah this is the Infinity thread. I was totally confused. I thought this was the supertask thread. Sorry for the confusion. Meta you did not hijack that thread. I can't believe they are still going on about that stupid lamp. I can't believe I get lost on this forum.

Meta, once I understood that Tones was arguing that set equality is the law of identity, I realized why you're arguing this point. I entirely agree with you. I apologize to you for jumping to multiple wrong conclusions.

I noticed that you posted then deleted a response to me, so perhaps you at first objected to my post then realized that by the end, I was in agreement with you. I'll think of it that way.

I haven't yet worked through Tones's reply to me outlining his argument, so I should reserve judgment. But at this moment it seems to me that set equality is a defined symbol in a particular axiomatic system. As such has no referent at all, any more than the chess bishop refers to Bishop Berkeley. It doesn't refer to anything concrete, nor anything abstract. It simply stands for a certain predicate in ZF. It can't possibly "know" about logic or metaphysics. It can't refer to "sets" since nobody knows what a set is. A set is whatever satisfies the axioms. And set equality is a relation between sets, which have no existence outside the axioms; and have no meaning even within the axioms.

Meta I agree with you on this point and had no idea that I was jumping into an ongoing conversation in a thread I didn't realize I was in. My bad all around. Tones, I look forward to working through your post. I'm sure many clever people must have considered this very problem of set equality and identity and there's a lot I don't know.

I didn't get the business about the Eleatics. Did they have this conversation back in the day?

I've moved your discussion on set ordering and the meaning of equality to this discussion — Michael

Explains a lot ...

There are three ways we could approach for set theory: — TonesInDeepFreeze

Ok I finally made a run at this, and I am having a little trouble understanding your meaning.

I believe you are trying to convince me that logical identity is the same thing as set equality as given by extensionality.

I'm a little unclear on what you mean by logical idenity. Do you mean the law of identity, everything is equal to itself? or identical to itself? Or did you mean something else?

I could use some specifics to help anchor my understanding.

(1) Take '=' from identity theory — TonesInDeepFreeze

I already got in trouble here! I looked up "identity theory." Both Wiki and IEP say it's a theory of mind. So that's not what you're talking about. Wiki has a disambiguation page that led me to Pure identity theories, a linkable paragraph within an article called List of first-order theories.

So if you could just define "identity theory" for me, and tell me what "=" means in that theory, I'll understand you better.

, with the axioms of identity theory, and add the axiom of extensionality. In that case, '=' is still undefined but we happen to have an additional axiom about it. — TonesInDeepFreeze

Even if I knew what you mean by identity theory, I still did not understand this. Still undefined but additional axiom. Sorry I don't follow. I'm probably missing your point I'm sure.

I have a question for you.

In ZF, I define $R = \{x \notin x\}$ and $S = \{x \notin x\}$, two definitions of the Russell set.

I ask: What is the truth value, if any, of $R = S$?

How should I think of this? In ZF? In set theories with classes?

The axiom of extensionality is not a definition there. And, with the usual semantics, '=' stands for the identity relation. It seems to me that this is the most common approach. — TonesInDeepFreeze

Ok, I hope I will understand this when you tell me what the identity relation is. But if extensionality is not an axiom, what is it? Axioms and definitions are the same thing. You can take them as "assumed true," or you can take them as definitional classifiers, separating the universe into things that satisfy the definition and things that don't.

For example, take the axioms for group theory. As axioms, they are assumed true for every group. But we actually use the axioms as a definition. If a mathematical structure obeys the axioms it's a group; and if not, not. So we can use axioms as a definitional boundary between everything we're interested in, and everything we're not. Axioms and definitions are the same thing viewed from different perspectives.

(2) Don't take '=' from identity theory.

Definition: x = y <-> Az(z e x <-> z e y)

Axiom: x = y -> Az(x e z -> y e z) — TonesInDeepFreeze

I don't really understand this. What are you trying to say in the axiom? That if two sets satisfy extensionality (the definition) then any set one of them is an element of, the other is also an element of? Am I getting that right? I think that already follows from the definition. In fact I convinced myself I could prove it, but did not work out the details. So I could be wrong about this.

But what is the intent?

(3) Don't take '=' from identity theory.

Definition: x = y <-> Az(z e x <-> y e z)

Axiom: Az(z e x <-> z e y) -> Az(x e z -> y e z)

With (2) and (3), yes, '=' could stand for an equivalence relation on the universe that is not the identity relation. But it seems to me that even in this case, we'd stipulate a semantics that requires that '=' stands for the identity relation. And I think it's safe to say that usually mathematicians still regard '1+1 = 2' to mean that '1+1' stands for the same number that '2' stands for, and not merely that they stand for members in some equivalence relation, and especially not that it's just all uninterpreted symbols. — TonesInDeepFreeze

I didn't get all this, what's the intent of the axiom, what does it all mean?

As far as 1 + 1 = 2, I've explained to @Metaphysician Undercover that these are two expressions that refer to the same set. By extensionality there is only one set, and two different representations of it. Other posters have mentioned that the intentional meanings of 1 + 1 and 2 are different, and the extensional meanings are the same. Mathematicians use the extentional meaning of a symbol. 1 + 1 = 2 "point" to the same abstract number in abstract number land, wherever that may be.

Mathematicians don't think of numbers as uninterpreted symbols. But perhaps if pressed to explain where numbers live, what kind of existence they have, it's safer to revert to formalism.

Suppose X and Y are objects in the universe, but they are not sets?
— fishfry



In set theory, contrary to a popular notion, we can define 'set':

x is a class <-> (x =0 or Ez z e x)

x is a set <-> (x is a class & Ez x e z)

x is a proper class <-> (x is a class & x is not a set)

x is an urelement <-> x is not a class — TonesInDeepFreeze

I don't know much about set theory with classes. I'm just a humble ZF guy. I have no choice! (set theory joke).

Then in ordinary set theory we have these theorems:

Ax x is a class

Ax x is set

Ax x is not a proper class

Ax x is not an urelement

If our meta-theory for doing models has only sets, then all members of universes are sets.

If our meta-theory for doing models has proper classes, still a universe is a set (proof is easy by the definition of 'model'). And no proper class is a member of a set.

If our meta-theory for doing models has urelements, and '=' stands for the identity relation, then the axiom of extensionality is false in any model that has two or more urelements in the universe or has the empty set and one or more urelements in the universe. — TonesInDeepFreeze

Didn't get to read much of this closely. Perhaps we can revisit later.



I did note one thing I disagreed with. You wrote:

"If our meta-theory for doing models has proper classes, still a universe is a set (proof is easy by the definition of 'model')"

Perhaps we're using different terminology. When they do independence proofs, models are sets. So for example to prove that ZF is consistent, we are required to produce a set that satisfies the axioms. It's no good to just provide a proper class, since the class of all sets satisfies the axioms but isn't sufficient to prove the consistency of ZF. (As usual I'm still confused about how ZF can see this, since it doesn't have proper classes).

Secondly, I know universes that are not sets. For example:

* The von Neumann universe and Gödel's constructible universe, both of which are proper classes (however you regard them) and are commonly called universes.

So perhaps I am not sure what is your definition of a universe.

Even informally, in ZF the universe is "all the sets there are." The axioms quantify over all the sets. And the universe of sets is not a set.

Summary of all this

These are some of my thoughts on reading your post. I'm sure I missed a lot. I'm still interested in understanding why you think that the logical identity (whatever that is, I'm still a little unclear) is the same thing as set equality under extensionality. I just can't see how that could even be. Once we get outside of the world of sets, we need a new definition. Because extensionality only applies to sets. And if we have a new definition, by definition we do not have the SAME definition. We have a DIFFERENT definition, which is to be consulted whenever we are wondering about the equality of two things that are not both sets.

It's like if I'm a computer multiplying integers, I use one algorithm. If I'm multiplying floats, I use another. So if we have two sets, we use extensionality to tell if A = B. If at least one of them is not a set, we have to use some OTHER way of telling. Which is your logical identity. It's a different thing, not the same thing.

Since I've convinced myself from first principles that logical identity and set equality are different things, I have a bit of a hard time following your arguments, since they must be wrong, or we must not be talking about the same thing.

Perhaps you can walk me through this slowly and clearly.

when we write:

x = y

we mean:

x is identical with y

x is equal to y

x equals y

x is y

x is the same as y

However, the poster, in all his crank glory, continues to not understand:

x = y

does NOT mean:

'x' is identical with 'y'

'x' is equal to 'y'

'x' equals 'y'

'x' is 'y'

'x' is the same as 'y'

but it DOES mean:

what 'x' stands for is identical with what 'y' stands for

what 'x' stands for is equal to what 'y' stands for

what 'x' stands for equals what 'y' stands for

what 'x' stands for is what 'y' stands for

what 'x' stands for is the same as what 'y' stands for — TonesInDeepFreeze

Just to humour you Tones, I read this post. So here's a question for you. When you state the law of identity as "a thing is identical with itself", would this identity include not only all of the thing's constituent elements, but also the ordering of those elements? For example, if I say that this rock is identical with itself, not only would this indicate that all the elements which compose the rock are identical with the rock, but also the ordering of those elements. If the ordering of the elements was not the same, then it would not be identical with the rock.

Meta, once I understood that Tones was arguing that set equality is the law of identity, I realized why you're arguing this point. I entirely agree with you. I apologize to you for jumping to multiple wrong conclusions. — fishfry

Apology accepted. I wrote most of the following before reading this, so ignore, or read, whatever suits you. The deleted post must have occurred in the transfer from the other thread. Michael was concerned about derailing the thread, and took out certain posts and transferred them.

I haven't yet worked through Tones's reply to me outlining his argument, so I should reserve judgment. But at this moment it seems to me that set equality is a defined symbol in a particular axiomatic system. As such has no referent at all, any more than the chess bishop refers to Bishop Berkeley. It doesn't refer to anything concrete, nor anything abstract. It simply stands for a certain predicate in ZF. It can't possibly "know" about logic or metaphysics. It can't refer to "sets" since nobody knows what a set is. A set is whatever satisfies the axioms. And set equality is a relation between sets, which have no existence outside the axioms; and have no meaning even within the axioms. — fishfry

As I indicate in my latest post in the supertask thread, Tones has a knack for taking highly specialized definitions designed for a particular axiomatic system, and applying them completely out of context. Be aware of that.

But nobody claims mathematical equality is identity. — fishfry

Tones does, obviously.

You would be fun in set theory class. You're entirely hung up on the very first axiom. "Class, Axiom 1 is the axiom of extensionality. It tells us when two sets are equal." You, three years later: "But that's not metaphysical identity! You mathematicians are bad people. And you don't understand anything!" And your professor goes, Meta, We still have a countable infinitely of axioms to get through! Can we please move on? — fishfry

I dropped out of abstract mathematics somewhere around trigonometry, for that very reason. I got hung up in my need to understand everything clearly, and could not get past what was supposed to be simple axioms. I had a similar but slightly different problem in physics. We learned how a wave was a disturbance in a substance, and got to play in wave tanks, using all different sorts of vibrations, to make various waves and interference patterns. Then we moved along to learn about light as a wave without a substance. Wait, what was the point about teaching us how waves are a feature of a substance?

But Tones is a bit different. Tones forges ahead with misunderstanding of fundamental axioms. Tones insists that the axiom of extensionality tells us when two sets are identical. He refers to something he calls "identity theory", which I haven't yet been able to decipher.

But axioms don'g mean anything. They're just rules in a formal game, like chess. As I say, if you asked a mathematician if mathematical equality is metaphysical identity, a few of them would have an educated opinion about the matter and they'd immediately agree with you. The rest, the vast majority, wouldn't understand the question and would be annoyed that you interrupted them. — fishfry

If axioms are rules, then they mean something. They dictate how the "formal game" is to be played. If the rules are misunderstood, as is the case with Tones, then the rules will not be properly applied.

You're fighting a straw man of your own creation. — fishfry

Tones is a monster, not of my own creation.

I remember fondly when I spent weeks trying to explain order theory to you, back when I thought you were trying to understand anything. You are still at this. If two sets have the same elements and the same order, they are equal as ordered sets. It's just about layers of abstraction, separating out concepts. First you have things, then you place them in order.

Somehow this offends you. Why? — fishfry

It is self-contradicting, what you say. " First you have things, then you place them in order."

If you have things, there is necessarily an order to those things which you have. To say "I have some things and there is no order to these things which I have, is contradictory, because to exist as "some things" is to have an order. Here we get to the bottom of things, the difference between having things, and imaginary things.

Yes I did actually understand that! I was just startled that Meta was still going on about order being an inseparable and inherent aspect of a set, when I had already had such a detailed conversation with him on this subject several years ago. I did actually realize you were quoting him -- I was just surprised to see him still hung up on that topic. — fishfry

You are taking Tones' misrepresentation. I fully respect, and have repeatedly told Tones, that sets have no inherent order, exactly as you explained to me, years ago. What I argue is that things, have an order to their constituent elements, and this is an essential aspect of a thing's identity. So I've been trying to explain to Tones, that the "identity" of a set (as derived from the axiom of extensionality) is not consistent with the identity of a thing (as stated in the law of identity). But Tones is in denial, and incessantly insists that set theory is based in the law of identity.

cc @TonesInDeepFreeze since your name got referenced a lot here.

Apology accepted. — Metaphysician Undercover

Thanks.

As I indicate in my latest post in the supertask thread, Tones has a knack for taking highly specialized definitions designed for a particular axiomatic system, and applying them completely out of context. Be aware of that. — Metaphysician Undercover

I often don't follow the purpose of his symbology.

You guys should stay out of that thread, you're not discussing supertasks and I can see why @Michael moved you.

But nobody claims mathematical equality is identity.
— fishfry

Tones does, obviously. — Metaphysician Undercover

Yes I didn't realize that. I don't see how it can be, but I'm not aware of how the logicians and set theorists resolve this.

I dropped out of abstract mathematics somewhere around trigonometry, for that very reason. I got hung up in my need to understand everything clearly, and could not get past what was supposed to be simple axioms. I had a similar but slightly different problem in physics. We learned how a wave was a disturbance in a substance, and got to play in wave tanks, using all different sorts of vibrations, to make various waves and interference patterns. Then we moved along to learn about light as a wave without a substance. Wait, what was the point about teaching us how waves are a feature of a substance? — Metaphysician Undercover

Yeah that business about waves without a medium is pretty murky. I've seen videos where they say, "It's a probability wave!" as if that explains anything. Probability waves are purely abstract mathematical gadgets, they aren't physical. Leaving unexplained the question of what electromagnetic waves are waving.

It's a well known problem that physics is no longer about the physical world, but rather about esoteric models that seem to work, without telling us much about the physical world. The "shut up and calculate" school of quantum physics.

But Tones is a bit different. Tones forges ahead with misunderstanding of fundamental axioms. Tones insists that the axiom of extensionality tells us when two sets are identical. He refers to something he calls "identity theory", which I haven't yet been able to decipher. — Metaphysician Undercover

Aha! I just asked him about this very point.

If axioms are rules, then they mean something. They dictate how the "formal game" is to be played. If the rules are misunderstood, as is the case with Tones, then the rules will not be properly applied. — Metaphysician Undercover

I can't comment on Tones's opinions in that area. I haven't been reading this thread.

Tones is a monster, not of my own creation. — Metaphysician Undercover

I can't tell if he knows a lot of logic but doesn't always explain himself, or is just typing stuff in. I had a very hard time with his last post to me explaining how identity was set equality.

It is self-contradicting, what you say. " First you have things, then you place them in order." — Metaphysician Undercover

No it's perfectly sensible. You have a class of screaming school kids, eight year olds say, on the playground. They're totally disordered. The only organizing principle is that you have a set of kids.

Then you tell them to line up by height. Now you have an ordered set of kids. Or you tell them to line up in alphabetical order of their last name. Now you have the same set with a different order.

It's an everyday commonplace fact that we can have a set of things in various orders.

Now maybe you are making the point that everything is in SOME order. The kids in the playground could still be ordered by their geographical locations or whatever.

But sets don't have inherent order.

If you have things, there is necessarily an order to those things which you have. To say "I have some things and there is no order to these things which I have, is contradictory, because to exist as "some things" is to have an order. Here we get to the bottom of things, the difference between having things, and imaginary things. — Metaphysician Undercover

Another way to look at it is that, as you say, perhaps every set has some inherent order, but we are just ignoring the order properties to call it a set. Then we bring in the order properties. It's just a way of abstracting things into layers.

But mathematical sets by themselves have no inherent order till we give them one. It's just part of the abstraction process.

You are taking Tones' misrepresentation. I fully respect, and have repeatedly told Tones, that sets have no inherent order, exactly as you explained to me, years ago. — Metaphysician Undercover

I'm moved that I had an effect. It was not in vain. I'm happy.

What I argue is that things, have an order to their constituent elements, and this is an essential aspect of a thing's identity. — Metaphysician Undercover

Yes, I am starting to come around to your point of view. But tell me this. Since, given a set, there are many different ways to order it, how do you know which one is inherently part of it?

So I've been trying to explain to Tones, that the "identity" of a set (as derived from the axiom of extensionality) is not consistent with the identity of a thing (as stated in the law of identity). — Metaphysician Undercover

I would go so far as to say that identity isn't set equality, because identity applies also to things that are not sets.

But if you ask me whether I think that two sets that are equal are identical, I'd have to say yes. Because if they're equal, they're the same set. Not because of metaphysics, but because of set theory. Set theory only talks about sets, and doesn't even say what they are. Nobody knows what sets are. They're fictional entities. They obey the axioms and that's all we can know about them.

But Tones is in denial, and incessantly insists that set theory is based in the law of identity. — Metaphysician Undercover

I can see that you've developed a bit of a, what is the word, obsession? attitude? annoyance? with him.

No it's perfectly sensible. You have a class of screaming school kids, eight year olds say, on the playground. They're totally disordered. The only organizing principle is that you have a set of kids. — fishfry

They're not totally disordered though. At any time you can state the position of each one relative to the others, and that's an order. When you say "they're totally disordered", that's just metaphoric, meaning that you haven't taken the time, or haven't the capacity, to determine the order which they are in.

Then you tell them to line up by height. Now you have an ordered set of kids. Or you tell them to line up in alphabetical order of their last name. Now you have the same set with a different order. — fishfry

Those are just 'identified orders'. When the kids are running free, in what we might call a 'random order', what you called "totally disordered", there is still an order to them, it has just not been identified. So, for the principle "height", we could make a map and show at any specific time, the relations of the tallest, second tallest, etc., and that would be their order by height. And we could do the same for alphabetic order. So we number them in the same way that you would number them in a line, first second third etc., then show with the map, the positions of first second third etc., and that is their order. The supposed "random order", or "totally disordered" condition, is simply an order which has not been identified.

Now maybe you are making the point that everything is in SOME order. The kids in the playground could still be ordered by their geographical locations or whatever.

But sets don't have inherent order. — fishfry

Yes, that's very apprehensive of you fishfry, and I commend you on this. Most TPF posters would persist in their opinion (in this case your claim of "totally disordered", which implies absolute lack of order), not willing to accept the possibility that perhaps they misspoke.

So that is the point, everything is in "SOME" order. Now, consider what it means to say "sets don't have inherent order". Would you agree that this sets them apart from real collections of things? A real collection of things, like the children, must have SOME order. And, this order which they do have, is very significant because it places limitations on their capacity to be ordered.

So when you said "first you have things, then you place them in order", we need to allow that the "things" being talked about, come to us in the first place, with an inherent order, and this inherent order restricts their capacity to be ordered. For example, let's say that the things being talked about are numbers. We might say that 1 is first, 2 is second, 3 is third, etc., and this is their "inherent order". This is the way we find these "things", how they come to us, 1 is synonymous with first, 2 is synonymous with second, etc., and that is their inherent order. The proposition of set theory, that there is no inherent order to a set, removes this inherent order, so we can no longer say that one means first, etc.. Now there cannot be any first, second, or anything like that, inherent within the meaning of the numbers themselves. This effectively removes meaning from the symbols, as you've been saying.

Another way to look at it is that, as you say, perhaps every set has some inherent order, but we are just ignoring the order properties to call it a set. Then we bring in the order properties. It's just a way of abstracting things into layers.

But mathematical sets by themselves have no inherent order till we give them one. It's just part of the abstraction process. — fishfry

Yes, I think I see this. I would say it's a type of formalism, the attempt to totally remove meaning from the symbols. The problem though, is that such attempts are impossible, and some meaning still remains, as hidden, and the fact that it is hidden allows it to be deceptive and misleading. So, by the abstraction process you refer to, we remove all meaning from the symbols, to have "no inherent order". Now, what differentiates "2" from "3"? They are different symbols, with different applicable rules. If what is symbolized by these two, can have "no inherent order", then the rules for what we can do with them cannot have anything to do with order. This allows absolute freedom as to how they may be ordered.

However, we can ask, can the two numbers,2 and 3, be equal? I don't think so. Therefore we can conclude that there actually is a rule concerning their order, and there actually is not absolute freedom as to how they can be ordered. The two symbols cannot have the same place in an order. Therefore, there actually is "SOME" inherent order to the set, a rule concerning an order which is impossible. And this is why I say that these attempts at formalism, to completely remove meaning which inheres within what is symbolized by the symbol itself, are misleading and deceptive. We simply assume that the formalism has been successful, and inherent meaning has been removed (we take what is claimed for granted without justification), and we continue under this assumption, with complete disregard for the possibility of problems which might pop up later, due to the incompleteness of the abstraction process which is assumed to be complete. Then when a problem does pop up, we are inclined to analyze the application as what is causing the problem, and the last thing we would do is look back for faults in the fundamental assumptions, as cause of the problem.

Yes, I am starting to come around to your point of view. But tell me this. Since, given a set, there are many different ways to order it, how do you know which one is inherently part of it? — fishfry

As described above, you need to look for what is inherent within the meaning of the symbol. Formalism attempts the perfect, "ideal" abstraction, as you say, which is to give the imagination complete freedom to make the symbol mean absolutely anything. However, there is always vestiges of meaning which remain, such as the one I showed, it is impossible that 2=3. The vestiges of meaning usually manifest as impossibilities. Any impossibility limits possibility, which denies the "ideal abstraction", by limiting freedom.

So to answer your question, the order which is inherent is not one of the orders you can give the set, it is a preexisting limitation to the orders which you can give. When we receive the items, what you express as "first we have the items", there is always something within the nature of the items themselves (what you call "SOME order"), as received, which restricts your freedom to order them in anyway whatsoever.

But if you ask me whether I think that two sets that are equal are identical, I'd have to say yes. Because if they're equal, they're the same set. Not because of metaphysics, but because of set theory. Set theory only talks about sets, and doesn't even say what they are. Nobody knows what sets are. They're fictional entities. They obey the axioms and that's all we can know about them. — fishfry

There are many different ways that "same" is used. You and I might both have "the same book". The word "set" used here is "the same word" as someone using "set" somewhere else. So it's like any other word of convenience, it derives a different meaning in a different sort of context. In common parlance, mathematicians might say "they are the same set", but I think that what it really means is that they have the same members. So that's really a qualified "same".

I can see that you've developed a bit of a, what is the word, obsession? attitude? annoyance? with him. — fishfry

Actually I got annoyed with Tones rapidly, when we first met, but now he just amuses me.

They're not totally disordered though. At any time you can state the position of each one relative to the others, and that's an order. When you say "they're totally disordered", that's just metaphoric, meaning that you haven't taken the time, or haven't the capacity, to determine the order which they are in. — Metaphysician Undercover

Ok. For things in the real world, they are already in some order, even if it's a complete state of disorder. Even a completely disordered collection of gas molecules in a container, at every instant each molecule is wherever it is. And that set of coordinates, locating every molecule in space, is the order.

I get that. But by the same token, there is no preferred order. Suppose for example that I got my schoolkids from the playground to line up single-file in order of height. And now YOU come along and say, "Ah, that is the inherent order, and all other orders are disorders of that."

But of course your observation was a complete accident. I could have lined them up alphabetically by last name.

So even among physical objects, if we allow that they are always in some order, even if it's disorderly; but nevertheless, there is no preferred or inherent order.

I believe you are saying there's an inherent order, have I got that right?

Those are just 'identified orders'. When the kids are running free, in what we might call a 'random order', what you called "totally disordered", there is still an order to them, it has just not been identified. — Metaphysician Undercover

Yes we see that the same way. Totally disordered gas molecules in a container, at every instant there is a list of all atoms and where they are in the box (more or less, quantum effects notwithstanding, but that's not the point I'm making). I'm agreeing with you that even the most disordered state is still an order. It's just the set of facts about where everything is.

So, for the principle "height", we could make a map and show at any specific time, the relations of the tallest, second tallest, etc., and that would be their order by height. And we could do the same for alphabetic order. So we number them in the same way that you would number them in a line, first second third etc., then show with the map, the positions of first second third etc., and that is their order. The supposed "random order", or "totally disordered" condition, is simply an order which has not been identified. — Metaphysician Undercover

Yes perfectly happy to regard whatever physical positions and attributes -- their state -- is regarded as the order they're in at that moment. I'm fine with that.

Yes, that's very apprehensive of you fishfry, and I commend you on this. Most TPF posters would persist in their opinion (in this case your claim of "totally disordered", which implies absolute lack of order), not willing to accept the possibility that perhaps they misspoke. — Metaphysician Undercover

You have indeed persuaded me that every collection of physical things has an order, even if they are apparently disordered. But sets, well you know ...

So that is the point, everything is in "SOME" order. — Metaphysician Undercover

Everything in the physical world.

Now, consider what it means to say "sets don't have inherent order". — Metaphysician Undercover

I have obviously spent much time considering that. As much time as I've spent explaining to you that sets don't have inherent order!

Would you agree that this sets them apart from real collections of things? — Metaphysician Undercover

Of course. Sets aren't real. They're a mathematical abstraction. I've never asserted otherwise.

A real collection of things, like the children, must have SOME order. And, this order which they do have, is very significant because it places limitations on their capacity to be ordered. — Metaphysician Undercover

Yes. I agree. But why does it matter? It doesn't apply to sets. You know why? Because that's what the axioms say. That's the ultimate source of truth. It's just axiomatics. I'm not sure why all this is important to you.

So when you said "first you have things, then you place them in order", we need to allow that the "things" being talked about, come to us in the first place, with an inherent order, — Metaphysician Undercover

No, I did not say that. I said that the things come to us WITHOUT an inherent order; and we place one on them.

And secondly, I was not talking about things in the world. I was talking about abstract mathematical structures. Objects of thought. Not of the world.

and this inherent order restricts their capacity to be ordered. — Metaphysician Undercover

Yeah it's harder to line the kids up by height depending on how they happen to be arranged on the playground. But sets aren't kids on the playground. Sets are a conceptual gadget for thinking, what would it be like to have a collection of things that don't have any order? What could we still say about them?

It's a philosophical game like that. And it's useful. It lets us separate out the facts about sets that depend on their order, from the facts that don't.

This is just the essence of mathematical abstraction.

For example, let's say that the things being talked about are numbers. We might say that 1 is first, 2 is second, 3 is third, etc., and this is their "inherent order". — Metaphysician Undercover

Well now that you mention it, no. 1, 2, 3, ... is NOT the inherent order of the set $\mathbb N$, believe it or not. On the other hand it sort of is, in a sneaky way. Von Neumann defined the symbols 1, 2, 3, ... in such a way that $n \in m$ if it happens to be the case that we want n < m to be true.

But we still have to (1) define what 1, 2, 3, ... are as set; and then (2) define that what we mean by n < m, is that $n \in m$ as sets.

So even though it's kind of rigged for 1, 2, 3, ... to be the natural order -- in fact that's what they call it, the natural order -- it is still explicitly defined. And without that definition, the natural numbers have no inherent order.

I know this is hard for normal humans to accept, since it's pretty obvious that 1 < 2 < 3 and so on. But mathematicians insist on being picky about how numbers and other things are defined. In the set-theoretic view of modern math, the numbers 1, 2, 3, ... are defined as particular sets, with no inherent order; and then we impose their order by leveraging the $\in$ operator.

Now I'm going to meet you halfway on something. I admitted that the set definitions of 1, 2, 3,... are already set up to leverage the $\in$. But you could say -- and I am going to agree with you -- that when John von Neumann invented the modern set-theoretic definition of the natural numbers; he already had a pre-intuition of the inherent order 1 < 2 < 3 ... and that's why he defined things to work out that way.

So even though formally we've removed the scaffolding and there's no inherent order in the finished mathematical theory; the inspiration for designing the theory that way was in fact the inherent order of the natural numbers, no matter what set theory says.

I am going to agree with you about that. Mathematics is mysteriously influenced by something "real" about even the most abstract things, like numbers. Formalism is defeated in the end. It's NOT just about the symbols. Math is expressing something real about the world.

Is any of this along the lines of your thinking?

This is the way we find these "things", how they come to us, 1 is synonymous with first, 2 is synonymous with second, etc., and that is their inherent order. — Metaphysician Undercover

I just talked myself into (almost) agreeing with what I take to be your point of view:

* Even though the numbers 1, 2, 3, ... have an inherent order; when we do set theory we pretend they don't, purely for the sake of the formalism. But the formalism is missing something important. The set-theoretic formalism denies that the numbers have an inherent order. But the counting numbers DO have an inherent order that is obvious to every school child. Therefore the set-theoretic natural numbers do not capture the full metaphysical properties of the natural numbers.

Have I got any of that right?

The proposition of set theory, that there is no inherent order to a set, removes this inherent order, so we can no longer say that one means first, etc.. — Metaphysician Undercover

Only to quickly put it back. And as I just acknowledge, the method of defining numbers as sets, and then being able to "define" the order 1 < 2 < 3 ... was obviously set up to facilitate just that. Showing that von Neumann had a priori knowledge of the order he was formalizing.

But that's not surprising, really. Even formalists don't think everyone's just making everything up. Math is "about" something "out there" in the world. Right? Am I making any sense to you?

Now there cannot be any first, second, or anything like that, inherent within the meaning of the numbers themselves. This effectively removes meaning from the symbols, as you've been saying. — Metaphysician Undercover

From the formal perspective of set theory. But not to deny that numbers don't have inherent order. Our formal model of numbers has no inherent order. But that's a virtue. It lets us study those aspects of sets that don't depend on order. I think I said that earlier. It's a process of abstraction, not lying for ill intent or metaphysical error.

Another way to look at it is that, as you say, perhaps every set has some inherent order, but we are just ignoring the order properties to call it a set. Then we bring in the order properties. It's just a way of abstracting things into layers. — Metaphysician Undercover

Yes if you'll accept that it's really all that's going on. Like when you go to the dermatologist to remove a mole, he doesn't send you in for a bunch of tests on your pancreas. He deals with one particular level of your entire being. He's not denying you have all these other organs. He's just focussing on one thing at a time.

It's perfectly ok to think of sets as having an inherent order and all their other possible orders, but we're just not concerned about them today. We only want to look at the property of membership.

Yes, I think I see this. I would say it's a type of formalism, the attempt to totally remove meaning from the symbols. — Metaphysician Undercover

Yes, so that we can reason precisely about the objects the symbols represent.

The problem though, is that such attempts are impossible, and some meaning still remains, as hidden, and the fact that it is hidden allows it to be deceptive and misleading. — Metaphysician Undercover

Maybe you mean that von Neuman secretly knew that 1 < 2 when he formalized the natural numbers in such a way to make 1 < 2 come out true.

But of course he did! So maybe you are getting at the intuition that guides mathematicians to do things the way they do.

In other words math is discovered inductively; and only presented deductively.

So, by the abstraction process you refer to, we remove all meaning from the symbols, to have "no inherent order". Now, what differentiates "2" from "3"? — Metaphysician Undercover

What differentiates any set from any other set? All together: The axiom of extensionality!

In von Neumann's clever encoding, we make the following symbolic definitions:

$0 = \emptyset$

$1 = \{0\}$

$2 = \{0, 1\}$

$3 = \{0, 1, 2\}$

and so forth. One virtue of this encoding is that the cardinality of each number is what it "should" be. The set representing 3 has cardinality 3. Which of course isn't within the theory, it's outside the theory. We secretly already know was 3 is even before we define it.

Is that one of the thing's you're getting at?

Anyway, back to the question. How do we know that 2 and 3 are not the same set?

Well $2 \in 3$, but $2 \notin 2$.

Therefore by extensionality, $2 \neq 3$, because they don't have exactly the same elements.

Perhaps you can begin to see the virtues of working a the set level separately from its order properties. We can see the mechanics of how to use the axiom of extensionality. No order properties are needed to determine that 2 and 3 are different sets. It's just a matter of ignoring hypotheses that you don't need for a particular argument.

Nobody is saying that a given set doesn't have an order, as well as a lot of other stuff. A topology, some algebraic operations, a manifold structure perhaps. But we can learn a lot just from restricting our attention to the membership relation and seeing what we can learn just about that.

They are different symbols, with different applicable rules. If what is symbolized by these two, can have "no inherent order", then the rules for what we can do with them cannot have anything to do with order. This allows absolute freedom as to how they may be ordered. — Metaphysician Undercover

Yes, math is often concerned with the most general kind of structure or conceptual framework.

What most people think of the order of the counting numbers: 1, 2, 3, ..., mathematicians call the "standard order" or "usual order," in contrast with many other interesting orders we could define.

While normal people only think about the usual order, mathematicians are people who think about all the different ways a set like the natural numbers can be ordered. In fact there's a beautiful order to the ways that the natural numbers can be well-ordered. Those are the ordinal numbers.

In other words the collection of all the ways a set can be ordered ... can itself have a natural order that we can study.

So that's the kind of thought process mathematicians enjoy, when they go from order to orders. From an order like 1 < 2 < 3, to the concept of order itself in its most abstract form.

However, we can ask, can the two numbers,2 and 3, be equal?
— Metaphysician Undercover

I believe I already demonstrated from first principles, from the ZF axioms, that 2 and 3 are not the same set; which, in the von Neumann encoding, proves that they are not the same number.

I don't think so. — Metaphysician Undercover

I just did. 3 contains an element, namely 2, that is not an element of 2. Therefore they're not the same set by the axiom of extensionality.

Therefore we can conclude that there actually is a rule concerning their order, and there actually is not absolute freedom as to how they can be ordered. — Metaphysician Undercover

I hope you have taken the foregoing to heart. We define 2 and 3 as particular sets, and by design they are different sets under extensionality. They do not have the same elements therefore they are not the same set. Under the identification of sets as representatives of numbers, they are not the same number.

The two symbols cannot have the same place in an order. — Metaphysician Undercover

I hope I've already convinced you that 2 and 3 are not the same number, and that we can demonstrate that using nothing more than the axiom of extensionality. In other words we do not need to use any order properties to prove that 2 and 3 are not the same number.

Therefore, there actually is "SOME" inherent order to the set, a rule concerning an order which is impossible. — Metaphysician Undercover

No. I proved $2 \neq 3$ using only their set membership properties and without any need to invoke their order properties, inherent or assigned.

And this is why I say that these attempts at formalism, to completely remove meaning which inheres within what is symbolized by the symbol itself, are misleading and deceptive. — Metaphysician Undercover

Then you have been proven wrong. I don't need to mention or consider or use any of the order properties of 2 and 3 to determine that they're different numbers.

We simply assume that the formalism has been successful, and inherent meaning has been removed (we take what is claimed for granted without justification), and we continue under this assumption, with complete disregard for the possibility of problems which might pop up later, due to the incompleteness of the abstraction process which is assumed to be complete. — Metaphysician Undercover

Entirely without rational basis. This para is a wild generalization of your complaint about 2 and 3, but I already showed how we can distinguish 2 and 3 using only their membership properties and not their order properties. If that was the basis for this paragraph, you have no basis. But even then, the para makes wild unsupported accusations.

You will need a much better example -- well an example, period -- of the formalism failing. It certainly held up to your first test. I used the axiom of extensionality to prove that 2 is not 3. And a good thing, too!

The set theoretic abstractions have held up for a century, from Zermelo's 1922 axiomitization to today.

Then when a problem does pop up, we are inclined to analyze the application as what is causing the problem, and the last thing we would do is look back for faults in the fundamental assumptions, as cause of the problem. — Metaphysician Undercover

You are thrashing away at a strawman you've created out of your imagination, and under the mistaken belief that we can't tell 2 from 3 without their order properties. But we can.

As described above, you need to look for what is inherent within the meaning of the symbol. Formalism attempts the perfect, "ideal" abstraction, as you say, which is to give the imagination complete freedom to make the symbol mean absolutely anything. However, there is always vestiges of meaning which remain, such as the one I showed, it is impossible that 2=3. The vestiges of meaning usually manifest as impossibilities. Any impossibility limits possibility, which denies the "ideal abstraction", by limiting freedom. — Metaphysician Undercover

You know you have stopped being clear and coherent in the last few paras. All based on a mistaken belief. This last para does not parse. Not for me anyway. Formalism is a tool, it's not the goal.

So to answer your question, the order which is inherent is not one of the orders you can give the set, it is a preexisting limitation to the orders which you can give. When we receive the items, what you express as "first we have the items", there is always something within the nature of the items themselves (what you call "SOME order"), as received, which restricts your freedom to order them in anyway whatsoever. — Metaphysician Undercover

That works for numbers. What about kids? What is the inherent order, the playground or the single file?

There are many different ways that "same" is used. You and I might both have "the same book". The word "set" used here is "the same word" as someone using "set" somewhere else. So it's like any other word of convenience, it derives a different meaning in a different sort of context. In common parlance, mathematicians might say "they are the same set", but I think that what it really means is that they have the same members. So that's really a qualified "same". — Metaphysician Undercover

No, you are consistently wrong about this. If A and B are sets and I can prove that A = B, then A and B are the same set. They are in fact the identical set, of which there is only one instance in the entire universe. They are NOT "two copies" or two distinct entities that we are calling the same by changing the meaning of the word "same."

"mathematicians might say "they are the same set", but I think that what it really means is that they have the same members."

DUH that is what it MEANS to be the same set. That is the ONLY thing it means to be the same set. Sets don't have any existence other that what the axioms say. There is nothing else to know about sets.

I can't believe you wrote that. Yes that is what it MEANS for two sets to be the same. That they have the same members. That's ALL it means and EVERYTHING it means.

You simply can't accept that and I don't know why. The knight in chess moves the way it does. Not for any reason other than those are the rules. Likewise two sets are the same when they have the same members. Period, end of story. That's it.

There is ONE SET {1, 2, 3}. That's the only one. There is exactly one instance of every set. If two supposed sets are equal they are the same sets. Like the morning star and the evening star, they're the same object. I don't know how many times I've explained this over the years and I can't understand why it eludes you or troubles you.

Actually I got annoyed with Tones rapidly, when we first met, but now he just amuses me. — Metaphysician Undercover

If you say so ...

I understood that Tones was arguing that set equality is the law of identity — fishfry

I did not say that.

I said that classical mathematics has the law of identity as an axiom and that classical mathematics abides by the law of identity.

t seems to me that set equality is a defined symbol in a particular axiomatic system. — fishfry

I addressed that. Written up in another way:

Ordinarily, set theory is formulated with first order logic with identity (aka 'identity theory') in which '=' is primitive not defined, and the only other primitive is 'e' ("is a member of").

But we can take a different approach in which we don't assume identity theory but instead define '='. I don't see that approach taken often.

But both approaches are equivalent in the sense that they result in the exact same set of theorems written with '=' and'e'.

I believe you are trying to convince me that logical identity is the same thing as set equality as given by extensionality. — fishfry

No, I am not saying any such thing.

(1) I don't think I used the locution 'logical identity'.

But maybe 'logical identity' means the law of identity and Leibniz's two principles.

Classical mathematics adheres to the law of identity and Leibniz's two principles.

The identity relation on a universe U is {<x x> | x e U}. Put informally, it's {<x y> | x is y}, which is {<x y> | x is identical with y}.

Identity theory (first order) is axiomatized:

Axiom:

Ax x = x (law of identity)

Axiom schema (I'm leaving out some technical details):

For any formula P(x):

Axy((P(x) & x = y) -> P(y)) (Leibniz's indiscernibility of identicals)


But identity theory, merely syntactically, can't require that '=' be interpreted as standing for the identity relation on the universe as opposed to standing for some other equivalence relation on the universe.

And Leibniz's identity of indiscernibles cannot be captured in first order unless there are only finitely many predicate symbols.

So we make the standard semantics for idenity theory require that '=' does stand for the identity relation. And then (I think this is correct:) the identity of indiscernibles holds as follows: Suppose members of the universe x and y agree on all predicates. Then they agree on the predicate '=', but then they are identical.

(2) The axiom of extensionality is a non-logical axiom, as it is true in some models for the language and false in other models for the language.

As mentioned, suppose we have identity theory. Then we add the axiom of extensionality. Then we still have all the theorems of identity theory and the standard semantics that interprets '=' as standing for the identity relation, but with axiom of extensionality, we have more theorems. The axiom of extensionality does not contradict identity theory and identity theory is still adhered to. All the axiom of extensionality does is add that a sufficient condition for x being identical with y is that x and y have the same members. That is not a logical statement, since it is not true for all interepretations of the language. Most saliently, the axiom of extensionality is false when there are at least two urelements in the domain.

In sum: Set theory adopts identity theory and the standard semantics for identity theory, and also the axiom of extensionality. With that, we get the theorem:

Axy(x = y <-> Az((z e x <-> z e y) & (x e z <-> y e z)))

And semantically we get that '=' stands for the identity relation.

And if we didn't base on identity theory, then we would have the above not as a non-definitional theorem but as a definition (definitional axiom) for '='; and we would still stipulate that we have use the standard semantics for '='.

Think of it this way: No matter what theory we have, if it is is built on identity theory, then the law of identity holds for that theory, and that applies to set theory in particular. But set theory, with its axiom of extensionality, has an additional requirement so that set theory is true only in models where having the same members is a sufficient condition for identity.

if you could just define "identity theory" for me, and tell me what "=" means in that theory — fishfry

As I said much earlier in this thread, it is the first order theory axiomatized by:

Axiom:

Ax x = x (law of identity)

Axiom schema (I'm leaving out some technical details):

For all formulas P(x):

Axy((P(x) & x = y) -> P(y)) (indiscernibility of identicals)

The meaning of '=' is given by semantics, and the standard semantics is that '=' maps to the identity relation on the universe. So, for any terms 't' and 's', 't = s' is true if and only if 't' stands for the same member of the universe that 's' stands for.

Still undefined but additional axiom. Sorry I don't follow. — fishfry

'=' is a primitive symbol. The axiom of extensionality is an additional axiom, not an axiom of identity theory.

In ZF, I define R={x∉x} — fishfry

Maybe you mean {x | ~ x e x} (you left out 'x |').

In Z we prove there is no set R such that Ax(x e R <-> ~ x e x). Therefore, the abstraction notation {x | ~ x e x} is not justified. How we handle that depends on our approach to abstraction notation. Personally, as a matter of style, I prefer the Fregean method, but reference-less abstraction notation is a whole other subject.

If extensionality is not an axiom, what is it? — fishfry

It is an axiom in ordinary set theory. I was describing a different approach, much less common, in which we don't have the axiom of extensionality.

Axioms and definitions are the same thing. You can take them as "assumed true," or you can take them as definitional classifiers, separating the universe into things that satisfy the definition and things that don't. — fishfry

There are two different senses:

(1) Syntactical definitions. These define symbols added to a language. In a theory, they are regarded as definitional axioms. But they are not like non-definitional axioms, in the sense that definitional axioms only provide for use of new symbols and don't add to the theory otherwise (criteria of eliminability and non-creativity).

This is the sense I'm using in my remarks about approaches to '=' in set theory.

(2) A set of axioms induces the class of models of the axioms. For example, we say first order group theory "defines" 'group'.

That is the sense that goes with the notion you mention

(2) Don't take '=' from identity theory.

Definition: x = y <-> Az(z e x <-> z e y)

Axiom: x = y -> Az(x e z -> y e z)
— TonesInDeepFreeze

If two sets satisfy extensionality (the definition) then any set one of them is an element of, the other is also an element of? — fishfry

Yes.

I think that already follows from the definition. In fact I convinced myself I could prove it, but did not work out the details. So I could be wrong about this. — fishfry

Do the details. Remember that you don't have the identity axioms, so you can't use anything prior about '='. For example, you can't use substitutivity.

But what is the intent? — fishfry

To fulfill the other approach where we don't start with identity theory.

(3) Don't take '=' from identity theory.

Definition: x = y <-> Az(z e x <-> y e z)

Axiom: Az(z e x <-> z e y) -> Az(x e z -> y e z)

With (2) and (3), yes, '=' could stand for an equivalence relation on the universe that is not the identity relation. But it seems to me that even in this case, we'd stipulate a semantics that requires that '=' stands for the identity relation. And I think it's safe to say that usually mathematicians still regard '1+1 = 2' to mean that '1+1' stands for the same number that '2' stands for, and not merely that they stand for members in some equivalence relation, and especially not that it's just all uninterpreted symbols.
— TonesInDeepFreeze

I didn't get all this, what's the intent of the axiom, what does it all mean? — fishfry

The intent is to arrive at the theorems of set theory but without adopting identity theory.

I did note one thing I disagreed with. You wrote:

"If our meta-theory for doing models has proper classes, still a universe is a set (proof is easy by the definition of 'model')"

Perhaps we're using different terminology. When they do independence proofs, models are sets. So for example to prove that ZF is consistent, we are required to produce a set that satisfies the axioms. It's no good to just provide a proper class — fishfry

What you quoted by me agrees with that. A universe for a model is a set. But that doesn't entail that our meta-theory cannot be a class theory, as long as universes for models are sets.

I know universes that are not sets. For example:

The von Neumann universe and Gödel's constructible universe, both of which are proper classes (however you regard them) and are commonly called universes. — fishfry

That's a different sense from 'universe for a model'.

your definition of a universe. — fishfry

A model M for a language is a pair <U F> such that U is a non-empty set and F is an interpretation function from the set of non-logical symbols. U is referred to as 'the universe for M'.

Even informally, in ZF the universe is "all the sets there are." The axioms quantify over all the sets. And the universe of sets is not a set. — fishfry

I'm not talking about informal usage such as that.

If set theory has a model (which we believe it does), then the universe for that model is a set. That universe doesn't have to be "all the sets" (which is an informal notion anyway). It's a purely technical point: A model for a language has a universe that is a set.

why you think that the logical identity (whatever that is, I'm still a little unclear) is the same thing as set equality under extensionality. — fishfry

I think no such thing.

/

Rather than sorting out your questions in this disparate manner, it would be better - a lot easier - to share a common reference such as one of the widely used textbooks in mathematical logic. I think Enderton's 'A Mathematical Introduction To Logic' is as good as can be found. And for set theory, his 'Elements Of Set Theory'.

I said that classical mathematics has the law of identity as an axiom and that classical mathematics abides by the law of identity. — TonesInDeepFreeze

You know, I studied math and I never heard that said, anywhere. I don't think it's true. I'd be glad for a mathematical reference. Pick up a text book on set theory and you won't find it.

I addressed that. Written up in another way:

Ordinarily, set theory is formulated with first order logic with identity (aka 'identity theory') in which '=' is primitive not defined, and the only other primitive is 'e' ("is a member of"). — TonesInDeepFreeze

I asked for a reference to "identity theory," since a Google search brings up many different meanings, none of them bearing on this topic.

But we can take a different approach in which we don't assume identity theory but instead define '='. I don't see that approach taken often. — TonesInDeepFreeze

Yes, there's a first-order theory of equality. Is that what you mean?

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

But both approaches are equivalent in the sense that they result in the exact same set of theorems written with '=' and'e'. — TonesInDeepFreeze

Maybe, I don't know. You haven't convinced me that set equality has anything to do with the law of identity.

No, I am not saying any such thing. — TonesInDeepFreeze

You seem to be saying that.

(1) I don't think I used the locution 'logical identity'.

But maybe 'logical identity' means the law of identity and Leibniz's two principles.

Classical mathematics adheres to the law of identity and Leibniz's two principles. — TonesInDeepFreeze

More rabbit holes. Nothing to do with set theory. I believe you have said logical identity. But if not, what do you mean? Define your terms please.

The identity relation on a universe U is {<x x> | x e U}. Put informally, it's {<x y> | x is y}, which is {<x y> | x is identical with y}.

Identity theory (first order) is axiomatized:

Axiom:

Ax x = x (law of identity)

Axiom schema (I'm leaving out some technical details):

For any formula P(x):

Axy((P(x) & x = y) -> P(y)) (Leibniz's indiscernibility of identicals) — TonesInDeepFreeze

None of those come up in set theory. Can you tell me, where did you come up with this line of discourse? Logic class? Logic in the philosophy department or the math department? Your own original ideas?

I was literally shocked the other day when you claimed that set equality is "identity," no matter how you define it. You still haven't made a case.

But identity theory, merely syntactically, can't require that '=' be interpreted as standing for the identity relation on the universe as opposed to standing for some other equivalence relation on the universe. — TonesInDeepFreeze

We're not having the same conversation now. I couldn't even parse that. It certainly doesn't tell me when two sets are equal.

And Leibniz's identity of indiscernibles cannot be captured in first order unless there are only finitely many predicate symbols. — TonesInDeepFreeze

You're just typing stuff in and not addressing the issue. How does set equality relate?

So we make the standard semantics for idenity theory require that '=' does stand for the identity relation. And then (I think this is correct:) the identity of indiscernibles holds as follows: Suppose members of the universe x and y agree on all predicates. Then they agree on the predicate '=', but then they are identical. — TonesInDeepFreeze

Losing me big time. I can't make sense of any of this in the context of set equality.

(2) The axiom of extensionality is a non-logical axiom, as it is true in some models for the language and false in other models for the language. — TonesInDeepFreeze

There is no model of set theory in which extensionality is false. None whatsoever.

As mentioned, suppose we have identity theory. Then we add the axiom of extensionality. Then we still have all the theorems of identity theory and the standard semantics that interprets '=' as standing for the identity relation, but with axiom of extensionality, we have more theorems. The axiom of extensionality does not contradict identity theory and identity theory is still adhered to. All the axiom of extensionality does is add that a sufficient condition for x being identical with y is that x and y have the same members. That is not a logical statement, since it is not true for all interepretations of the language. Most saliently, the axiom of extensionality is false when there are at least two urelements in the domain. — TonesInDeepFreeze

I don't believe that, but I don't know anything about sets with urelements. Have a reference by any chance? Or give me an example.

In sum: Set theory adopts identity theory and the standard semantics for identity theory, and also the axiom of extensionality. With that, we get the theorem:

Axy(x = y <-> Az((z e x <-> z e y) & (x e z <-> y e z)))

And semantically we get that '=' stands for the identity relation. — TonesInDeepFreeze

Can't possibly be.

And if we didn't base on identity theory, then we would have the above not as a non-definitional theorem but as a definition (definitional axiom) for '='; and we would still stipulate that we have use the standard semantics for '='. — TonesInDeepFreeze

You speak a funny language. What subject is this? Where did you learn this? Have you a reference?

Think of it this way: No matter what theory we have, if it is is built on identity theory, then the law of identity holds for that theory, and that applies to set theory in particular. But set theory, with its axiom of extensionality, has an additional requirement so that set theory is true only in models where having the same members is a sufficient condition for identity. — TonesInDeepFreeze

Nonsense. Models of what? How do you know whether having the same members is sufficient for identity unless you already have the axiom of extensionality? In which case you're doing set theory, not identity theory.

Rather than sorting out your questions in this disparate manner, it would be better - a lot easier - to share a common reference such as one of the widely used textbooks in mathematical logic. I think Enderton's 'A Mathematical Introduction To Logic' is as good as can be found. And for set theory, his 'Elements Of Set Theory'. — TonesInDeepFreeze

I'll stipulate that maybe you have a point to make. You have not communicated it to me. I didn't have the heart to tackle this long post tonight.

(edit) I see this was a response to my earlier long post, so I really should read this. I skimmed a bit. I'm sure you have something in mind. You could even be right. Maybe I'll get to this at some point.

I checked my copy of Kunen, Set Theory: An Introduction to Independence Proofs. It's a standard graduate text in set theory.

He states that "=" is one of the symbols of the theory, and that "informally it stands for equality." He doesn't go any deeper in that direction. So if someone else has explained these matters more deeply, and if that person is Enderton, I'll have to take your word for it.

From your previous post, I wonder if you can give an example of a set theory with urelements in which extensionality doesn't apply.

And for set theory, his 'Elements Of Set Theory'. — TonesInDeepFreeze

I found a pdf of that here:

https://docs.ufpr.br/~hoefel/ensino/CM304_CompleMat_PE3/livros/Enderton_Elements%20of%20set%20theory_%281977%29.pdf

I looked at it and he starts with the axiom of extensionality, which he states in the conventional manner, without reference to identity.

Please let me know what page I should look at in this book to see any kind of discussion of the issues you bring up. I'll wait for your response before looking for a copy of his logic book, because I can skim a set theory book and have an idea what's going on, but not so for logic.

Classical mathematics is regarded as being formalized by ZFC. ZFC starts with a base of first order logic with identity. Whether called 'first order logic with identity', or 'first order logic with equality', or 'identity theory', the usual axioms, whether named as axioms for 'identity' or axioms for 'equality' are as I mentioned. For example, Enderton's 'A Mathematical Introduction To Logic'.

One of the axioms of identity theory is the law of identity formalized: Ax x=x. So the law of identity pertains to ZFC and, if ZFC is consistent, then ZFC does not contradict identity theory.

Ask anyone who studies set theory, whether ZFC is a first order theory with identity. I can't help that the Google doesn't help to find this.

The Wikipedia article you mentioned is not well written. (1) It doesn't give an axiomatization and (2) It doesn't mention that we can have other symbols in the signature and that by a schema we can generalize beyond a signature with only '='.

But, yes, of course we can use the word 'equality' or the word 'identity', as I said so many times. Perhaps 'equality' is used more often, but the exact formal theory is the same, and is axiomatized as I mentioned.

You haven't convinced me that set equality has anything to do with the law of identity. — fishfry

I gave you fulsome explication. Said yet another way:

The axiom of extensionality gives a sufficient condition for equality. But it doesn't give a necessary condition. So it is not a mathematical definition. A formal mathematical definition for a binary predicate R is of the form:

x R y if and only if (something here about x and y)

A definition of '=' would be of the form:

x = y if and only if (something about x and y)

But the axiom of extensionality is usually given:

If, for all z, z is a member of x if and only if z is a member of y, then x= y.

That is of the form:

If (something about x and y) then x = y. And that is only a sufficient condition, not a biconditional.

But I also mentioned that we could have this variation:

x = y if and only if (for all z, z is a member of x if and only if z is a member of y; and for all z, x is a member of z if and only if y is a member of z).

And that would be a definition, since it gives both a sufficient and necessary condition for x = y.

However, I'll say it yet another way:

Ordinarily, set theory is written with a signature of '=' and 'e', where '=' is logical and 'e' is non-logical, and we have the axioms for first order logic with identity (aka 'idenity theory' or 'first order logic with equality'). Then we add the axiom of extensionality. And then we get (1) as a theorem.

No matter which approach we take, we end up with the same theorems.

you claimed that set equality is "identity," — fishfry

For the third time, I did not say that. And I again told you what I do say. Please stop saying that I said something that I did not say.

I said that set theory adheres to the law of identity but that set theory, with its axiom of extensionality, adds an additional sufficient condition for identity.

I don't know why you don't grasp this:

Ax x= x is an axiom of first order logic with identity. And set theory is a theory that subsumes first order logic with identity. So Ax x=x is also an axiom incorporated into set theory. But also set theory adds the axiom of extensionality.

You seem to be saying that. — fishfry

You are egregiously glossing over what I exactly say. So you form incorrect "seems".

If, for whatever reason, there is a point of my terminology that requires definition for you, then, time permitting, I would supply the definition. Or if there is an argument you can't see to be logical, then, time permitting, I would explain it in even more detail if there even is more detail that could be reasonably provided. However, that can lead to a long chain of definitions back to primitive notions, so it would be better to start at the beginning such as in some chosen textbook. But that does not justify you claiming that I said things that I did not say.

But identity theory, merely syntactically, can't require that '=' be interpreted as standing for the identity relation on the universe as opposed to standing for some other equivalence relation on the universe.
— TonesInDeepFreeze

We're not having the same conversation now. I couldn't even parse that. — fishfry

It parses perfectly even if it seems difficult when one is not familiar with the basic mathematical logic in which symbols are interpreted with models. I'll put it this way:

'=' is a 2-place predicate symbol.

A model (an interpretation) for a language assigns a 2-place relation on the universe to a 2-place relation symbol. In other words, that is an assignment of the meaning, per the model, of the 2-place relation symbol. We call that 'the interpretation of the symbol'. That is semantical.

In general, relation symbols are interpreted differently with different models. But in the special case of '=', we stipulate that, with all models, '=' is stands for the identity relation on the universe for the model.

So my point was that from the mere syntactical presence of '=' in a formula, we can't ensure that '=' stands for the identity relation, and we have to turn to semantics (models) for that.

There is no model of set theory in which extensionality is false. — fishfry

Correct. There are no models of the axiom of extensionality in which the axiom of extensionality is false. But there are models in which the axiom of extensionality is false; they are not models of the axiom of extensionality. Take this is steps:

A model M is for a language.

Theories are written in languages.

So if a theory T is written in a language L, then a model for M for L is model for the language of T.

Given a model M, some sentences in the language L are true in M and some sentences in the language L are false in M.

A theory is a set of sentences closed under provability.

If every sentence in theory T is true in a given model M, then we say "M is a model of T".

So, notice that that there is a difference in meaning between "M is a model for the language L" and "M is a model of the theory T".

That is a crucial thing to understand and keep in mind.

Now, back to the axiom of extensionality.

Let T be any theory that is axiomatized with a set of axioms that includes the axiom of extensionality. If M is a model of T, then the axiom of extensionality is true in M.

But there are models for the language for set theory that are not models of set theory. For example

Let M have universe U = {0 1} and let 'e' be interpreted as the empty relation.

M is a model for the language for set theory, and M is not a model of set theory.

Again, to stress:

There is a difference between a language and a theory writtten in that language.

Let L be a language, and T be a theory written in L, and M be a model for L. It is not entailed that M is a model of T.

/

Urelements. Even though search engines are often deficient, I bet that you can find articles about urelements.

Df. x is an urelement <-> (~Ey yex & ~ x = 0) ('0' here standing for 'the empty sety')

A theorem of Z:

~Ex x is an urelement.

But we could have other axioms where

~Ex x is an urlement

is not a theorem.

And we could have axioms where

Ex x is an urlement

is a theorem.

But a theory that has the theorem:

Exy (x is an urelement & y is an urelement & ~ x = y)

is obviously inconsistent with the axiom of extensionality.

But we could do this:

Axiom: Axy((~ x is an urlement & ~ y is an urelement & Az(z e x <-> z e y)) -> x = y)

In sum: Set theory adopts identity theory and the standard semantics for identity theory, and also the axiom of extensionality. With that, we get the theorem:

Axy(x = y <-> Az((z e x <-> z e y) & (x e z <-> y e z)))

And semantically we get that '=' stands for the identity relation.
— TonesInDeepFreeze

Can't possibly be. — fishfry

It be's. Just as I've explained again. And I'll explain yet another way:

Certain proofs, including in set theory, use the identity axioms I mentioned. Set theory also has the axiom of extensionality, which allows for even more proofs. And we have this theorem of set theory:

Axy(x = y <-> Az((z e x <-> z e y) & (x e z <-> y e z)))

That's all purely syntactical.

Meanwhile, we interpret '=' to stand for the identity relation on the universe of any model for the language for set theory. And that is semantical.

Models of what? — fishfry

Models for the language of set theory. Some of them are models of set theory.

Always keep in mind the distinction between "model M for a language L" and "model M of a theory T" (or, mentioning the language also, "model M of a theory T written in a language L").

How do you know whether having the same members is sufficient for identity unless you already have the axiom of extensionality? In which case you're doing set theory, not identity theory. — fishfry

You're very mixed up thinking that you're somehow disagreeing with me on those points.

Indeed, I am saying that the axiom of extensionality is what makes Az(z e x <-> z e y) sufficient for x = y.

And, indeed, identity theory is not set theory. Rather, identity theory is a sub-theory of set theory:

Every theorem of identity theory is a theorem of set theory. But not every theorem of set theory is a theorem of identity theory: Right off the bat, the axiom of extensionality is not a theorem of identity theory.

And again, we needn't quibble that "pure" identity theory does not have 'e' in its signature. I'm talking about the axioms of identity theory written in a signature that includes 'e'.

The Wikipedia article you mentioned is not well written. (1) It doesn't give an axiomatization and (2) It doesn't mention that we can have other symbols in the signature and that by a schema we can generalize beyond a signature with only '='. — TonesInDeepFreeze

I'm only trying to figure out what you're talking about. There are many theories of identity. Wiki actually has a disambiguation page on the subject. Give me a reference to this identity theory you keep talking about.

You pointed me to Enderton. I pulled a pdf of his set theory book and found nothing beyond the standard explication of the axiom of extensionality, with no reference to "identity" in any context.

I gave you the pdf. Please tell me what page to read to understand your point.

Or if it's in his logic book, give me a reference for that.

I appreciate that you wrote me another lengthy post, but I'm about two or three lengthy posts behind you, and it would be infinitely helpful if, having said that it's all in Enderton, if you'd give me a page reference in his set theory book; and failing that, his logic book.

I can't deal with the rest of this now. You said Enderton, I pulled Enderton. Give me a page ref please. His statement of extensionality is just like everyone else's and absolutely nothing like yours.

So Ax x=x is also an axiom incorporated into set theory. — TonesInDeepFreeze

Not in Enderton. Not in Kunen (grad level). Not in Halmos. Not in any set theory book I know, though I don't know many. Used to have a copy of Shoenfield but that was a long time ago and I don't remember what he said on the subject.

The Enderton reference was to the identity axioms. See page 112 in the logic book. And also, on page 83, he specifies satisfaction regarding '=' so that it adheres to interpreting '=' as the identity relation.

For set theory, an example is Hindman's 'Fundamentals Of Mathematical Logic' in which he is explicit that set theory is a first order theory. And earlier in the book, he gives the logical axioms as including the axioms regarding '=' similarly to the way I did. And he also mentions that the interpretation of '=' is the identity relation.

In general, even if many texts don't belabor that set theory is a first order theory, surely you don't dispute that it is? And that is first order theory with identity. To demonstrate:

Suppose you have only the axioms of set theory and no axioms for identity, then how do you suppose you would derive:

Axyz((x = y & y =z) -> x = z)

Watch out: You can't use "substitution of equals for equals", since that principle is derived from the identity axioms.

More generally, let P be a formula with only x free. How would you derive?:

(P(x) & x = y) -> P(y)

For example:

How would you derive?:

(x is finite & x = y) -> y is finite

Basically, the axiom of extensionality lets you infer equality, but it usually doesn't help in inferring from equality.

Yes, from x and y having the same members, we can infer that x is y. But from "x is y", and without identity axioms, how would we infer very much else?

You need axioms for '=' other than the axiom of extensionality to deduce all the theorems of set theory. And those other axioms are axioms of identity, such as the axiomatization I've been mentioning. You need the identity axioms and the axiom of extensionality to get all the theorems.

Sure, in informal expositions, even as found in many set theory textbooks, we don't belabor our use of the axioms and theorems of identity theory, but take them as implicit in our arguments, especially since the principles (such as substitution of equals for equals) are engrained in mathematical reasoning. But when we rigorously formalize, we need the identity axioms. Meanwhile, such a book as Hinman, which is quite scrupulous does mention the identity axioms explicitly as among the first order axioms and does mention that set theory is a first order theory.

Also: That textbooks might not mention something, especially as textbooks often don't belabor every formal detail (and some hardly even glance upon even basic formal considerations), it is not entailed that we can't specify details that are left out. And even if terminology used is not common (such as, 'identity theory' seems less common than 'first order logic with equality') we should allow the terminology as long as it is defined. And I did exactly define it, as I defined it as the theory axiomatized by the exact axioms I specified.

I have an idea that may help pedagogically.

In discussions about languages, models and theories, the prepositions 'for' and 'of' might get overlooked if one is not reading closely. But we can eschew those prepositions:

language
model (aka 'interpretation' or 'structure' (different from another sense of 'structure' in mathematics))
theory

are key concepts in mathematical logic.

We can stipulate this terminology:

M interprets a language L iff M is a pair <U F> where U is a non-empty universe and F is an interpretation function from the non-logical symbols of L

A theory T is written in a language L

M models a theory T iff every sentence in T is true in M

Then we have:

There are languages L, models M and theories T such that: T is written in L, and M interprets L, but M does not model T.