@fishfry
From that wiki page:

"or in words:

Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B."

I see the phrase "A is equal to B", but where does it indicate that A is the same as B?

From that wiki page:

"or in words:

Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B."

I see the phrase "A is equal to B", but where does it indicate that A is the same as B? — Metaphysician Undercover

It's better for me if I defer continuing this discussion at this time.

But for what it's worth, the symbol string "same" has no meaning in ZF. I do not know what it means, and I do not need to know what it means in order to do set theory.

The symbol "=" is being defined by the axiom of extensionality. You're adding things that aren't in the game. It's as if I'm trying to teach you chess and you say, "Where are the zebras?"

Ha, ha, very funny. — Metaphysician Undercover

I wasn't being funny.

But for what it's worth, the symbol string "same" has no meaning in ZF. — fishfry

"Same" has a meaning in the law of identity. So when you say that the axiom of extensionality is a statement of "identity", you are employing the concept of "same", where it does not belong. "Same" is implied by "identity". That's why I argued that to interpret the axiom of extensionality in this way, as a statement of identity, is a faulty interpretation.

You're adding things that aren't in the game. — fishfry

Actually, it is you who is adding things that aren't in the game, with your faulty interpretation. You are adding "identity", when the law of extensionality is really a definition of "equal". As I've been telling you, equality and identity are not the same concept. This is because "identity" implies "the same" whereas "equal" does not. So, we can have two different things which are equal, but two different things cannot have the same identity.

Therefore it is incorrect to interpret the axiom of extensionality, which is clearly an expression of equality, as an expression of identity. To interpret as a statement of identity is to add something which is not in the game, sameness, when the statement really concerns equality instead, which does not imply "same".

Set theoretic axioms can be difficult to anyone, so let's think about this.

What do you think identity in mathematics / set theory is?

So the axiom extensionality is that sets are equal if they have the same elements, if I understand it correctly.

So I think then the question for you, @Metaphysician Undercover, is how is the identity different between two sets that have the same elements?

Because you say "to read the axiom of extensionality as indicating identity rather than as indicating equality is a misinterpretation", it seems that you think this is different. A lay person would think that a set defined by it's elements.

And please just look how identity is defined in mathematics, and you'll notice what @fishfry is talking about.

__ __ __

Nice to see you, @fishfry on the forum again! It's been a while. :grin:

I don't prefer Wikipedia as a reference on such matters, but it was asked where in the Wikipedia article on the 'Axiom of extensionality' is it said that 'equals' means 'the same'.

The article states that the axiom of extensionality uses '=' with regard to predicate logic, with a link to an article on 'First-order logic'. And that article correctly states that the most common convention is that 'equals' means 'the same'. Moreover, the article on 'Equality (mathematics)' defines equality as sameness, and the article on the equals sign refers to equality, and the article on 'Identity (mathematics)' refers to equality.

In ordinary contexts in mathematics, including mathematical logic, including set theory, 'equals' means 'is the same as', which means the same as 'is identical with'. This is formalized by identity theory, which extends first order logic without identity, and adds a primitive binary predicate symbol '=' with axioms and a semantics.

More specifically:

/

Identity theory is first order logic plus:

Axiom: Ax x=x

Axiom schema:
For all formulas P,
Axy((x=y & P(x)) -> P(y))

Semantics:

For every model M, for all terms T and S,
T = S
is true if and only if M assigns T and S to the same member of the universe.

/

Set theory can be developed in at least two ways:

(1) First adopt identity theory. This gives us:

Theorem: Axy(x=y -> Az((xez <-> yez) & (zex <-> zey)))

Then add the axiom of extensionality:

Axiom: Axy(Az(zex <-> zey) -> x=y)

This gives us:

Theorem: Axy(x=y <-> Az(zex -> zey))

Thus, with identity theory and the axiom of extensionality, every model of
Az(zeT <-> zeS)
is a model that assigns T and S to the same member of the universe.

(2) Don't adopt identity theory. Instead:

Definition: Axy(x=y <-> Az((xez <-> yez) & (zex <-> zey)))

That gives us as theorems all the axioms of identity theory.

However, if we don't also stipulate the semantics of identity theory, the axioms of identity theory along with the axiom of extensionality do not provide that every model in which S=T is true is a model that assigns S and T to same member of the universe.

Nice to see you, fishfry on the forum again! It's been a while. — ssu

Thank you. This forum drives me to extended vacations sometimes.

You are adding "identity" — Metaphysician Undercover

I'm pretty sure I never said that, but if I did, please supply a reference to my quote.

when the law of extensionality is really a definition of "equal". — Metaphysician Undercover

Now you're getting it.

With identity theory, '=' is primitive and not defined, and the axiom of extensionality merely provides a sufficient basis for equality that is not in identity theory. Without identity theory, for a definition of '=' we need not just the axiom of extensionality but also the 'xez <-> yez' clause.

What do you think identity in mathematics / set theory is? — ssu

I don't think mathematics/set theory deals with identity at all. I think that identity is an ontological principle defined by the law of identity. Mathematics deals with equality, which is distinct from identity. Some people on this forum have argued against this, claiming that there is an identity within set theory. But then they only seem to be able to argue that "equal" means "the same", when clearly this is false.

So I think then the question for you, Metaphysician Undercover, is how is the identity different between two sets that have the same elements? — ssu

Having the same elements does not mean being the same as. Having a different order for example would make two sets with the same elements not "the same" by the law of identity which indicates that a thing is the same as itself only. So, even identifying them as "two sets" indicates that they have a different identity.

Because you say "to read the axiom of extensionality as indicating identity rather than as indicating equality is a misinterpretation", it seems that you think this is different. A lay person would think that a set defined by it's elements. — ssu

Yes, identity is very different from equal. By the law of identity, we only call it "the same" if it is one and the same thing. The computer I typed on this morning is the same computer as the one I type on now. Two equal things are not necessarily the same. There were other computers at the store which are equal to mine but each one of them is different, i.e. not the same computer as the one I brought home. I think a lay person would agree with that.

And please just look how identity is defined in mathematics, and you'll notice what fishfry is talking about. — ssu

If you can find that definition for me, I'll take a look. Then we can discuss whether "identity" in mathematics is consistent with the law of identity.

I'm pretty sure I never said that, but if I did, please supply a reference to my quote. — fishfry

My apologies, for misrepresenting what we argued about. I thought you argued that the axiom of extensionality indicated identity.

Again, as has been mentioned very many times on this forum, the use of the symbol '=' and the words 'equal' and 'identical' in mathematics are by stipulation. By use of such stipulations we do not claim that the words are used exactly as they are used in all the very many other different contexts and senses in everyday language and in philosophy. This kind of thing should not have to be pointed out so very many times in a philosophy forum.

As to sets and order, as has been demonstrated very many times on this forum, sets with at least two members have different orderings, so there is not "the" ordering of a set.

A while ago, I gave this example: The set whose members are all and only the bandmates in The Beatles is a set. But there is not "the" ordering of that 4 member set. Indeed there are 24 orderings of that set:

https://thephilosophyforum.com/discussion/comment/884421

Or put it this way, if every set has an order that is "the" order of the set, then the set whose members are all and only the bandmates in The Beatles has an order that is "the" order. If one will venture to state which of the 24 orders of that set is "the" order, then I can ensure that we could find at least 23 Beatles fans who would disagree with that being "the" order.

A definition of 'identity' was requested and the poster said he will look at it. In identity theory in mathematics, '=' is not primitive. But the semantics require that S=T is true if and only if 'S' and 'T' name the same thing. To look at this in more detail and with all the groundwork for it provided, one may look one of many introductory textbooks in mathematical logic.

Again, whatever "the axiom of extensionality indicates identity means":

(1) If we use identity theory at the base of set theory, then the axiom of extensionality merely adds a sufficient condition for '='. And the semantics of identity theory provide that '=' means 'the same as' or 'is identical with'.

(2) If we do not use identity theory at the base of set theory, then then we may use the axiom of extensionality but augmented with an additional clause to define '='. However, without the semantics of identity theory, it is not the case that such an axiom alone proves that '=' means 'the same as' or 'is identical with'.

My apologies, for misrepresenting what we argued about. I thought you argued that the axiom of extensionality indicated identity. — Metaphysician Undercover

Apology accepted. I do see how my view may have seemed that way to you. For example I am certain that I'd have maintained that if A and B are sets, and we can write A = B, then A and B are the same set.

That is certainly true in set theory. But I think it's really more true in the metatheory or the way we talk about set theory, than set theory itself.

In set theory, "same" is a shorthand for "satisfies the premises of the axiom of extensionality." You are trying to overload the word with metaphysical baggage that it simply does not have in math. The axiom of extentionality is syntax. You are imbuing it with semantics that you are making up or bringing over from other meanings of the word you may know. You need to take things on their own terms when studying any technical field.

What I mean is, formal set theory says:

"If such-and-so, then we can write A = B."

In casual, everyday talk about set theory, we say, "A and B are the same set if they have exactly the same elements."

So you see there is a gap between those. Set theory is a purely syntactic exercise. If, given the definition of "=" as in the axiom of extensionality, we can derive a formal proof from the axioms that A = B, then we can write A = B from now on.

But all this talk of "sameness" is really a very loose casual adaptation of the axiom of extensionality. And in so doing, we seem to add semantics to it. As if we are making a metaphysical claim that A = B.

But in actuality we are not doing that!! Rather, we're simply claiming that the symbol "=" is to henceforth be defined as this other condition.

So the set theory is syntactic; and it's a mistake to confuse our everyday casual talk about set theory, with some kind of ontological claim.

tl;dr: When a set theorist says two sets are "the same," there is a formal derivation from first principles that A and B satisfy the premises of the axiom of extensionality. It's a purely syntactic exercise.

They are NOT implying any kind of metaphysical baggage for the word "same." If pressed, they'd retreat to the formal syntax.

Make sense? You are using "same" with metaphysical meaning. Set theorists use "same" as a casual shorthand for the condition expressed by the axiom of extensionality. It's a synonym by definition. The set theorist's "same" is a casual synonym; your "same" is some kind of ontological commitment. So all this is just confusion about two different meanings of the same word.


Also, meta: This thread, "Infinity," is active, and I keep getting mentions for it and replying. But this thread does not show up in my front-page feed! Anyone seeing this or know what's going on?

You are trying to overload the word with metaphysical baggage that it simply does not have in math. — fishfry

What I've argued is that this use of "same", is not consistent with "same" as defined by the law of identity. And if this sense of "same" is claimed to be constitutive of "identity", as Tones argues, then this is a violation of the law of identity. If this is what you want to call "metaphysical baggage", that's ok with me. There are many words with similar forms of "metaphysical baggage". The use of a specific word in one field may contradict its use in another field, and it only becomes a problem if people start to believe that the two uses are consistent with one another.

They are NOT implying any kind of metaphysical baggage for the word "same." If pressed, they'd retreat to the formal syntax. — fishfry

You may want to take this up with Tones, and his notion of "identity theory", which obviously is a kind of metaphysical baggage.

Make sense? You are using "same" with metaphysical meaning. Set theorists use "same" as a casual shorthand for the condition expressed by the axiom of extensionality. It's a synonym by definition. The set theorist's "same" is a casual synonym; your "same" is some kind of ontological commitment. So all this is just confusion about two different meanings of the same word. — fishfry

That completely makes sense. However, not every mathematician is as reasonable as you are. If you look at what TonesoffTheDeepEnd is writing here, you'll see great effort to support some kind of formal identity theory. That is not a "casual shorthand for the condition expressed by the axiom of extensionality".

Also, meta: This thread, "Infinity," is active, and I keep getting mentions for it and replying. But this thread does not show up in my front-page feed! Anyone seeing this or know what's going on? — fishfry

What has happened is that there was a new policy initiated which was to relegate a whole lot of these rambling, bickering, blah. blah, blah, type of threads to The Lounge. And threads in The Lounge don't show on the front page. You'll find The Lounge in the list of Categories on the left side of the front page.

I didn't say anything about 'constitutive'.

And it is exactly my point that use of terminologies in different fields are often not compatible with one another, and, as I have said many times in this forum, and again in this thread, mathematics makes no claim that '=', 'equals' and 'is identical with' are used in mathematics in the same senses as in all those in everyday life and in other fields of study.

And I don't have a personal sense of 'identity theory'. I am merely referring to a publicly studied formal theory.

And I don't claim to "support" identity theory. I am merely saying what it is, what some of its theorems are, something about the semantics that goes with it, and how it relates in certain ways to set theory.

There is nothing "off the deep end" about anything I've said here. Barely clever putdowns by means of renaming posters might be at least minimally apropos if they were based on at least something.

Meanwhile, I'm still interested in hearing what one would claim to be "the" order of the set of all and only the bandmates in The Beatles.

That is just one of myriad examples. Without an answer, the notion that every set has its "the" ordering is dead in the water.

This is telling:

The poster challenged by asking where in a certain Wikipedia article it says that 'equals' means 'the same'. I pointed out: The article states that the axiom of extensionality uses '=' with regard to predicate logic, with a link to an article on 'First-order logic'. And that article correctly states that the most common convention is that 'equals' means 'the same'. Moreover, the article on 'Equality (mathematics)' defines equality as sameness, and the article on the equals sign refers to equality, and the article on 'Identity (mathematics)' refers to equality.

I don't usually reference Wikipedia, but at least it is abundantly clear that the poster's challenge regarding what Wikipedia does happen to say is answered, when he could have found out it out for himself. It's typical of the poster to stick with his method of railroading full speed ahead with his own claims and challenges while hardly ever granting that they have been answered.

If '=' in set theory is to mean 'is the same as', it is not the case that the treatment of identity in set theory can dispense semantics.

Again, usually set theory presupposes identity theory, in which case it is by semantics that the interpretation of '=' is stipulated, and in which case '=' means 'is the same as'. And if set theory does not presuppose identity theory, then the axiom of extensionality is not enough syntactically, as we need the axiom of extensionality with an added clause. And still that is not enough to have that '=' means 'is the same as'. The details were given here:

https://thephilosophyforum.com/discussion/comment/897006

Regarding placement of threads: Some of the moderation of this forum is quite irrational.

I don't think mathematics/set theory deals with identity at all. — Metaphysician Undercover

If you can find that definition for me, I'll take a look. Then we can discuss whether "identity" in mathematics is consistent with the law of identity. — Metaphysician Undercover

Ok,

Even if the discussion has moved on, I'll just point out this, what identity in math is and why math does deal with identity:

In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables within a certain range of validity.[1] In other words, A = B is an identity if A and B define the same functions, and an identity is an equality between functions that are differently defined.

or in an other way:

An identity is an equation that is true for all values of the variables. For example:

(x+y)2 = (x2+2xy+y2)

The above equation is true for all possible values of x and y, so it is called an identity.

And what it isn't:

An identity is true for any value of the variable, but an equation is not. For example the equation
3x = 12
is true only when x=4, so it is an equation, but not an identity.

Also, meta: This thread, "Infinity," is active, and I keep getting mentions for it and replying. But this thread does not show up in my front-page feed! Anyone seeing this or know what's going on? — fishfry

It's in the Lounge.

It was deemed not Philosophical enough, or just math. Or lousy math. :yikes:

The reason is that the Ukraine crisis thread and the Israel-Palestine thread (Israel killing civilians in Gaza and the West Bank) along other political threads got so heated and ugly, the admins decided to put them into the Lounge (meaning not Philosophical debates). Having to do with the appearance that a Philosophy Forum site would discuss eloquently Philsophy, I guess. :snicker:

Even if the discussion has moved on, I'll just point out this, what identity in math is and why math does deal with identity:

In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables within a certain range of validity.[1] In other words, A = B is an identity if A and B define the same functions, and an identity is an equality between functions that are differently defined. — ssu

All right, you just confirmed what I thought. "Identity" in mathematics is equality. That clearly violates the law of identity. The law of identity allows that identity is a relation between a thing and itself, so there is only one thing involved. Equality on the other hand allows that two distinct things may be equal to each other. So unless you can provide this distinction between equality and identity, or else show that to be equal necessarily means to be one and the same thing, in set theory, then you ought to accept this violation. And, as I mentioned already, we know that the latter cannot be done, because set theory allows that two sets with the same elements in different orders are equal. Therefore I think we ought to just conclude that "identity" in mathematics is a violation of the law of identity. Agree?

"Identity" in mathematics is equality. — Metaphysician Undercover

A certain kind of equality, identity is an equation that us true for all values of its variables.

An equation might be true for some variables, like x+1=3 is true if (iff?) x=2. There's also equality, but not identity. Hence equality isn't always identity.

Law of identity, that each thing is identical with itself, isn't actually math, but general philosophy. So I guess the law of identity is simply a=a or 1=1. Yet math it's actually crucial to compare mathematical objects to other (or all other) mathematical objects. Hence defining a set "ssu" by saying "ssu" = "ssu" doesn't say much if anything. Hence the usual equations c=a+b.

It's very hard to think here that math would go against logic, so this is more of a mixture of definitions here. It's like comparing what in Physics is work and what in economics / sociology is work. The definitions are totally different.

The law of identity is a philosophical principle.

It is adopted in mathematics.

Ax x=x
is math.

/

Using '=', 'equals', and 'is identical with' interchangeably does not violate the law of identity.

Suppose I owe a creditor a certain amount of money, and ask them, "I have record of my balance as being 582 dollars plus 37 dollars. Do you have the same number?" They say, "Yes, your balance is 619 dollars and 0 cents." It would be ridiculous for me to say, "No! 582 plus 37 is not the same number as 619.00!"

582+37 is the same number as 619.00.

582+37 is identical with 619.00.

582+37 is equal to 619.00.

582+37 = 619.00.

That is not vitiated by the fact that:

'582+37' is not the same expression as '619.00'

Even a child can understand that

2+2 = 4 means that '2+2' and '4' name the same number.

/

In set theory, there are no two different sets with the same elements and different orderings.

However, for any set with at least two elements, there are different orderings on that set. To express a set S with an ordering R on S and a different ordering Y on S, we may simply say:

R is an ordering on S & Y is an ordering on S & ~R=Y

To talk about a set S and a particular ordering R on S we may mention;

<S R>

Millions of people who have studied mathematics understand that. Including those who have built the digital computers we are using at this moment.

/

There are 24 orderings of the set whose members are the bandmates in the Beatles. That doesn't entail that there is more than one set whose members are the bandmates in the Beatles. There is only one such set. It is the set whose members are the bandmates in the Beatles no matter how you order them.

So, I'm still curious what "the" ordering of the Beatles is supposed to be.

If one cannot answer that, then one ought not claim that for every set there is "the" order of that set.

And, in this particular case, by a lack of response to the question, I take it that the poster who makes that claim has no answer.

On a philosophy forum, one of its most prolific posters cannot fathom the use-mention distinction.

"To the Lounge with this rubbish" indeed!

t's in the Lounge.

It was deemed not Philosophical enough, or just math. Or lousy math. — ssu

Thanks.

Having to do with the appearance that a Philosophy Forum site would discuss eloquently Philsophy, I guess. — ssu

I'm sure that might happen someday .... /s

That completely makes sense. However, not every mathematician is as reasonable as you are. If you look at what TonesoffTheDeepEnd is writing here, you'll see great effort to support some kind of formal identity theory. That is not a "casual shorthand for the condition expressed by the axiom of extensionality". — Metaphysician Undercover

I'm not a mathematician. I studied math in school, long ago.

Thanks for calling me reasonable.

I can't defend the views of other posters, and I can't engage with what someone else might have said.

Regarding placement of threads: Some of the moderation of this forum is quite irrational. — TonesInDeepFreeze

For sure. The mods have an aversion to math-related topics that I don't understand.

Law of identity, that each thing is identical with itself, isn't actually math, but general philosophy. So I guess the law of identity is simply a=a or 1=1. Yet math it's actually crucial to compare mathematical objects to other (or all other) mathematical objects. Hence defining a set "ssu" by saying "ssu" = "ssu" doesn't say much if anything. Hence the usual equations c=a+b. — ssu

This is the crucial point, which I'd like to bring to your attention. The law of identity is an ontological principle which deals which "objects" as we meet them in our daily lives, all the different objects which we sense around us, and it forms a defining principle of what it means to be an object. It is meant to recognize the reality of these objects, in the face of skepticism, and offer guidance as to the type of existence which they have. It is probably the most universal of general principles, applying to all the different types of objects that we might possibly encounter.

On the other hand, supposed "mathematical objects" are distinctly different, and the law of identity does not apply. Now we might simply consider that the so-called mathematical objects are not even objects, therefore they don't have any identity as such, and they are really just thoughts and ideas. That seems like a reasonable approach. However, some people want to compose a special type of "identity", specifically designed for these thoughts and ideas, and through reference to this special type of identity they argue that these thoughts and ideas actually are a special type of object, mathematical objects.

However, it ought to be clear to you, that this is just smoke and mirrors sophistry. The special type of identity is formulated for the special purpose of creating the illusion that these thoughts and ideas are a type of object. But it is actually impossible that these ideas are a type of object because the law of identity applies to all types of objects which we might possibly encounter, and this supposed special type of object requires a distinct form of identity which is incompatible with the law of identity.

It's like comparing what in Physics is work and what in economics / sociology is work. The definitions are totally different. — ssu

I don't think that's true true, "work" in physics is consistent with "work" in economics/sociology, physics is simply a broader sense, and "work" in sociology is narrowed down to work is done by human beings.

In the case of "identity" in mathematics, it is inconsistent with, (in violation of), "identity" in the law of identity.

Suppose I owe a creditor a certain amount of money, and ask them, "I have record of my balance as being 582 dollars plus 37 dollars. Do you have the same number?" They say, "Yes, your balance is 619 dollars and 0 cents." It would be ridiculous for me to say, "No! 582 plus 37 is not the same number as 619.00!" — TonesInDeepFreeze

Obviously, you do not understand the law of identity. By the law of identity, the symbols printed here, 582, are distinctly different from the symbols printed over here, 582. Although they appear very similar there is a different position, context, etc., so it is very clear that this is not two instances of one and the same thing, 582. By the law of identity two separate occurrences of the printed symbols, are not the same thing, they are two similar things. Under the conditions of the law of identity we cannot say that 582 printed here is the same thing as, or has the same identity as, 582 printed here.

Therefore I recommend that any wise person would completely disregard the following statement, as coming from the mind of someone who does not know what they are talking about.

The law of identity is a philosophical principle.

It is adopted in mathematics. — TonesInDeepFreeze