There's a point from a while back. Maybe we can fix it.

I said that

ExAy y e x

is consistent.

You disputed that.

So I pointed out that I am not saying it is consistent with set theory, which has the axiom schema of separation from which we derive:

~ExAy yex.

Rather, it is consistent just as it stands alone.

I said "It is consistent onto itself." Yet, you still disputed me. Much later it dawned on me that you were thinking that I meant 'onto' as with a surjection. But I meant 'onto itself' to mean 'in and of itself'. And later I found out that people don't usually say 'onto itself' that way. So I saw that I had lapsed in English.

So here we are, and I am hoping that you see that I was correct that

ExAy yex

is consistent in and of itself, even though not consistent with the axiom schema of separation.

At the time I proved by adducing this model:

U = {0}

'e' stands for {<0 0>}

I need to go from x = y to saying that for all z, x in x iff z in y. — fishfry

A typo there? I think you meant 'z in x iff z in y'?

Without sarcasm I say that it gives me a good feeling that reason, intellectual curiosity and communication have won the day finally. — TonesInDeepFreeze

Yes. Quite the epiphany. I've actually just found several web pages and articles explaining all this. One even mentioned that the converse of extensionality follows from Leibniz (either ident of indisc. or other way 'round). Evidently I'm the last person to find this out. Even the Wiki page on extensionality mentions this, and I thought I'd read it several times but evidently not that part.

Late here way past bedtime I actually need to be somewhere tomorrow morning I'm going to regret this. Will be offline till tomorrow evening or day after.. Thanks for the insight. It was the proof of the transitive property that did it. Once I realized I needed the converse, the floodgates opened. Great example.

Ok more later. Thanks again.

We can state the indiscernibility of identicals as a first order schema, no matter how many nonlogical symbols there are in the language.

And we can state the identity of indiscernibles as a first order schema if there are only finitely many nonlogical symbols in the language.

But it's interesting that we cannot state the identity of indiscernibles as a first order schema if there are infinitely many nonlogical symbols in the language.

If you mean that it would help for my posts to link to yours, then I'll hope not to forget doing that each time. — TonesInDeepFreeze

Not link, quote. Either quote a fragment of my post, as I just did to yours; or else just mention me as @fishfry, where you have to type "" around the handle name.

Linking posts is something else, at the bottom you can get a hard link to the post, but you don't need to do that.

My preference regarding you is that you don't gloss my posts and jump to conclusions that I've said something I didn't say but that you think I must have said in you own confusions or lack of familiarity with the concepts or terminology. — TonesInDeepFreeze

Something about the scorpion and the frog. You expect me to stop having the many flaws I have? I will do my best, but interact with me at your discretion.

I have a large pile of mentions, so I'll get to them and save yours for later. I'm still in the afterglow of my set theoretic epiphany. I understood your point. You're right and I was confused, but now, thanks to your untiring efforts, you have unconfused me. Actually I think I was just hallucinating, because I do know that extensionality is an implication and not a bi-implication. I just never thought about the converse. But the converse is the "portal to the next level down," predicate logic.

I'm happy to have clarified this, it makes a lot of sense.

I think we can jump forward past the extensionality. The moment I saw the problem with proving the transitivity of set equality, I was enlightened. I swear, I almost literally smacked my head. "I can't use extensionality. I need the converse. So you picked the perfect puzzle to get through to me.

So going forward, I stand educated on this point. And although I do try my best not to exhibit my flaws, well, I may yet leap to an unwarranted conclusion now and then.

I am hopelessly behind composing posts in at least a few threads. Even years behind in threads that I just had to let go because I really should be spending my time on other things more important than posting. — TonesInDeepFreeze

Shouldn't we all!

I wasn't clear; I didn't mean a URL link; I meant a reply link. Does the link in this post do what you want?

@fishfry

Now that we got the axiom of extensionality straightened out, it's apropos to get the rest of the dissension worked out.

It starts with these good posts:

https://thephilosophyforum.com/discussion/comment/911857

https://thephilosophyforum.com/discussion/comment/913150

The crank asked about rocks. But we were not talking about rocks. We have been talking about sets. Sets of rocks, or set of numbers, etc. Sets have orderings, but if a set has more than one members then it has more one ordering. For example, a set with two members:

{0 1} = {1 0}

There are two orderings of that set:

{<0 1>} and {<1 0>}

So there is not "THE" ordering of that set, since there are two of orderings of the set.

But we may indicate the set with regards to a particular ordering. The notation is:

<S R> where S is the set and R is a particular ordering. For example"

<{0 1} {<1 0>}>

is the set {0 1} along with the ordering that is the greater-than relation on the set.

For example, the set whose members are all and only the bandmates in the Beatles has 24 orderings. So there is not "THE" ordering of that set.

But we may indicate that set with regards to a particular ordering. For example, the alphabetical ordering by first name:

{<George Harrison, John Lennon> <George Harrison, Paul McCartney> <George Harrison, Ringo Starr> <John Lennon, Paul McCartney>, <John Lennon, Ringo Starr> <Paul McCartney, Ringo Starr>}

As a sequence: {<1 George Harrison> <2 John Lennon> <3 Paul McCartney> <4 Ringo Starr>}

As a list: George Harrison, John Lennon, Paul McCartney, Ringo Starr.

But, obviously there are many other ways to order the Beatles: by age from youngest to oldest, by age from oldest to youngest, by height from tallest to shortest, by height from shortest to tallest, by wealth number of record sales as an artist after the Beatles, ...

So there is not "THE" ordering of the set whose members are the bandmates in the Beatles.

But what about that rock? If it's the one that is the crank's head, then it is indeed empty and there is only one ordering of the set of its particles, which is the empty ordering.

But what about more complicated, more intelligent rocks? The rock is not a set. However, we may speak of the set of particles of the rock. And in that case, again, there is no "THE" ordering of that set. But the crank mentions structure. Yes, we may describe the rock in terms of a certain structure. But the rock, even as described per a certain structure is not a set; it's a rock. Moreover, we may describe a rock as different isomorphic structures. Your structure is based on rock's pointy tip facing up, and my structure, isomorphic to your structure is based on the rock's pointy tip facing down.

/

The crank says I use definitions out of context. The crank confuses self-description with outward observation.

/

The crank says that he doesn't know what I mean by 'identity theory' even though I've stated and explained the axioms of identity theory at least a few times. (Or if I hadn't done that prior to the crank's post, then nothing was stopping him from asking me to do it.)

/

The crank makes the ridiculous claim that I misunderstand the rules of axiom systems. I understand the formation syntax of the formal languages, the formation syntax of the formulas, the formation syntax of the axioms, the formation syntax of the inference rules - all recursively. And the formation of the semantics for the meaning of the formulas - all by inductive definition. I understand exactly how to check that a purported formal proof is a proof and also I understand exactly how to interpret the meaning of formulas.

The crank doesn't know what he's talking about regarding mathematics or the axiomatic method or regarding me. Then he says that I annoyed him when we met but now I merely amuse him. Ah, the classic arch line, "You merely amuse me". The crank is not only a feeble thinker, he's a lame flamer. And why was he initially annoyed? Because as he was freely spewing confusion, ignorance and disinformation on this forum, I corrected him.

/

The crank repeats his argument that the notion of identity in mathematics is wrong since mathematics regards objects that don't exist. So, yet again, the crank just ignores the responses I've given to that. Just to start: He ignores even the examples I've given of sets of non-abstract objects, such as the set of pencils strewn on my desk, etc.

doesn't always explain himself, or is just typing stuff in. — fishfry

I explain in detail. And it's a stupid thing to say that I just type stuff. But in post or even a series of them, I can't fit in an explanation all the way back to the basics of the subject, so if one doesn't have the benefit of a context of adequate knowledge, it's not my fault that I can't supply all that needed context in even several posts.

Then you tell [the kids on the playground] 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. — fishfry

Exactly and well put. I've given the crank that same explanation. He will never understand it, because he wants to not understand it. If he found himself understanding it one day, then he would face the crisis of seeing that he's been confused and in the dark for years and years (decades?).

The crank's rejoinder is that we may state the positions and that that is "an order".

He is exactly right there. It is AN order. He said it himself! It is not "THE" order since there are different orders, each of them AN order.

HALFTIME COMMENTARY: For those viewers who might wonder if this thread analogizes everyday discussions in mathematics among its various practitioners, let me assure it does not - at least from my antiquated perspective. Expertise in the "finer" points of logic is rarely required in traditional math, although,I admit, I've lost track of the enormous varieties of the subject over the passing years.

And perhaps I am wrong: checking ArXiv.org I see that in the past week there have been around 25 new logic papers submitted - about the same number as those in my area, complex analysis. And the axiom of extentionality on Wikipedia garners about 60 views per day - a healthy enough following.

Just passing thoughts when reflecting on the current discussion. Kudos to the three or four involved. :clap:

But what about that rock? If it's the one that is the crank's head, then it is indeed empty and there is only one ordering of the set of its particles, which is the empty ordering. — TonesInDeepFreeze

Excellent, I love it. TPF's head sophist has a sense of humour.

The elements of sets have no inherent order. — fishfry

OK, so here we have the issue. Remove the examples of real world objects (schoolkids etc.) as "the elements", and what exactly is an element? It cannot be a particular thing, because it does not obey the law of identity, so it is some sort of universal, an abstraction. But what type of abstraction is it, one which we pretend is a particular? Why is it pretended that these are particulars? Maybe so that the set can be subjected to bijection, and have cardinality. The question then is whether the elements are truly individuals, or just pretend individuals.

Sets have no meaning whatsoever, other than that they obey the axioms of set theory. — fishfry

Isn't that exactly what meaning is, obeyance of some rules? Now, we know what a set is, something which obeys the rules of set theory, the real issue though is what is an element of a set.

This was in response to your denial of the empty set. Tell me exactly -- and be extremely clear and specific, please -- tell me what other rule of set theory is contradicted by the empty set. — fishfry

It seems you are having problems understanding the inherent difficulty of the empty set. I think we'd better have clear agreement on what an element is before we approach that more difficult problem of the empty set.

I have explained to you the ontology of sets many times. They are mathematical abstractions. — fishfry

Yes, but you also claim that sets have no meaning. An abstraction with no meaning is contradictory. That's why I can't understand your teachings about set theory.

You know, I am not sure I agree that sets are universals. My understanding is that "fish" is a universal, and the particular tuna that ended up in this particular can of tuna I bought at the store today is a particular instance of the category or class of fish.

Sets are not like that at all.

I did ask you a long time ago to explain what you meant by universals, and you snarked off at me. And now you come back at me claiming that sets are universals. Explain to me what you mean by that.

The concept of a set is a universal. The set of rational numbers is a particular set, of which there is exactly one instance. — fishfry

Any abstraction is a universal because its applicable to more than one particular set of circumstances. Whatever it is that any multitude of particulars has in common, is a universal.

You appear to be suggesting a third category other than particular and universal, an abstraction which is not a universal. Care to explain?

LOL. Oh man you're crackin' me up. The set of rational numbers most definitely has a cardinality of ℵ0
ℵ
0
, because of Cantor's discovery of a bijection between the rational numbers and the natural numbers. — fishfry

Bijection is a problem, because it requires that the elements are individuals, particulars, which I argue they are not. This is why we need to clear up, and agree upon the ontological status of an "element" before we proceed.

Exactly and well put. I've given the crank that same explanation. He will never understand it, because he wants to not understand it. If he found himself understanding it one day, then he would face the crisis of seeing that he's been confused and in the dark for years and years (decades?). — TonesInDeepFreeze

I suggest we adhere to the principle you stated, the elements of a set are not things, like schoolkids, rocks or anything else. TPF's head sophist doesn't respect this principle.

...the set whose members are all and only the bandmates in the Beatles... — TonesInDeepFreeze

However, we may speak of the set of particles of the rock... — TonesInDeepFreeze

Etc..

The crank says, "TPF's head sophist has a sense of humour."

So the sophist crank finally comes close to a true sentence, but still only half true. I'm not a sophist, neither philosophically nor rhetorically.

The sophist crank says, "the principle you stated, the elements of a set are not things".

I never said any such thing. I've said the opposite. The sophist crank again lies about me, as a function of his abysmal confusion.

The sophist crank says, as a paragraph, "Etc.."

More eloquent than a rock, by a word.

OK, so here we have the issue. Remove the examples of real world objects (schoolkids etc.) as "the elements", and what exactly is an element? — Metaphysician Undercover

In general, excepting the somewhat lesser-known example of set theories with urlements, the elements are other sets. If we are justified, given the axioms (whichever we choose) of set theory, to write:

$x \in y$

then we may colloquially read this as, "x is an element of y." That's what an element is.

It cannot be a particular thing, because it does not obey the law of identity, so it is some sort of universal, an abstraction. — Metaphysician Undercover

Actually I am wrong about that @TonesInDeepFreeze showed me the error of my ways. All sets satisfy the law of identity. If I have a set X, I may write X = X by way of the law of identity. I do not need the axiom of extensionality for that. Perfectly clear to me now.

But what type of abstraction is it, one which we pretend is a particular? — Metaphysician Undercover

The law of identity applies to sets. So this line of argument is null and void.

Why is it pretended that these are particulars? Maybe so that the set can be subjected to bijection, and have cardinality. The question then is whether the elements are truly individuals, or just pretend individuals. — Metaphysician Undercover

If your criterion is that they satisfy the law of identity, they do. So your concern is addressed.

Isn't that exactly what meaning is, obeyance of some rules? — Metaphysician Undercover

Yes, very good. A group is any mathematical structure that obeys the axioms for groups. A set is any mathematical object that obeys the axioms for sets.

Now, we know what a set is, something which obeys the rules of set theory, the real issue though is what is an element of a set. — Metaphysician Undercover

Typically it's another set. Sets are subject to the law of identity. This should satisfy your concerns.

It seems you are having problems understanding the inherent difficulty of the empty set. — Metaphysician Undercover

I believe in the field of psychology, this is known as projection. YOU have problems with the empty set. I have no such problems. The empty set is the set of purple flying elephants in my left pocket. Oh wait you don't like "real life" examples. Never mind.

The empty set is the set of things that violate the law of identity. In symbols:

$\emptyset = \{x : x \neq x \}$

Happy now? (Of course you're not!) There are other formulations.

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

I think we'd better have clear agreement on what an element is before we approach that more difficult problem of the empty set. — Metaphysician Undercover

An element of a set is a the left side of an expression $x \in y$ that can be deduced from the axioms of set theory. x is the element, and y is a set. But x is typically a set as well. Think paper bags inside of paper bags. Oops there I go with real world analogies again.

Yes, but you also claim that sets have no meaning. — Metaphysician Undercover

They can be viewed that way from a formalist perspective.

It's of no importance to set theory. Certainly sets don't necessarily have real-world referents, since sets are quite a bit stranger than paper bags or collections in general.

What of it?

An abstraction with no meaning is contradictory. That's why I can't understand your teachings about set theory. — Metaphysician Undercover

What does chess mean?

See https://plato.stanford.edu/entries/abstract-objects/ and tell me if you find anything interesting in there.

Any abstraction is a universal because its applicable to more than one particular set of circumstances. Whatever it is that any multitude of particulars has in common, is a universal. — Metaphysician Undercover

Do you see the difference between the concept of set, and the concept of the set {1, 2, 3}?

One's a general set, and the other's a particular set.

Since you won't define a universal in such a way that you can sort this terminology out, I think your idea of universals must be vacuous. Fish is to this particular tuna on the end of my fishing line, as sets are to the set {1,2,3}. There is nothing problematic about that.

You appear to be suggesting a third category other than particular and universal, an abstraction which is not a universal. Care to explain? — Metaphysician Undercover

Me? I'm making no such suggestion.

Bijection is a problem, because it requires that the elements are individuals, particulars, — Metaphysician Undercover

They are, as far as I understand your use of the terminology, which you refuse to explain.

which I argue they are not. This is why we need to clear up, and agree upon the ontological status of an "element" before we proceed. — Metaphysician Undercover

An element is a set in a set theory without urelements. We say x is an element of y if we can legally write $x \in y$. Nothing could be simpler.

Here is another real world example.

Fairy tale characters are an abstract universal. They are general, and they don't actually exist.

Cinderella is a particular fairy tale character. She doesn't exist either, but she is an INSTANCE of the category of fairy tale characters.

Fairy tale characters are abstract universals, and Cinderella is an abstract particular.

In your world you don't have any abstraction at all. I think you're taking a point too far.

I explain in detail. And it's a stupid thing to say that I just type stuff. But in post or even a series of them, I can't fit in an explanation all the way back to the basics of the subject, so if one doesn't have the benefit of a context of adequate knowledge, it's not my fault that I can't supply all that needed context in even several posts. — TonesInDeepFreeze

I would say that your communication style, with me at least, tends to be confusing. The only thing you wrote that made sense to me was the challenge to prove the transitivity of set equality. Once I realized I needed the converse of extensionality, I was enlightened.

Many other things you wrote were lost on me. I know this frustrates you, but it's like fishing. You had to go through a whole container of worms to finally hook the fish(fry). You should be happy, instead of complaining about the wasted worms.

I apologize for the typing things in remark. I must have written that before I understood your point.

Now that we got the axiom of extensionality straightened out, it's apropos to get the rest of the dissension worked out.

It starts with these good posts: — TonesInDeepFreeze

What dissension? I'm happy I understood your point. I prefer not to go back into the old posts.

I wasn't clear; I didn't mean a URL link; I meant a reply link. Does the link in this post do what you want? — TonesInDeepFreeze

Yes, point being that if I'm away from the board for a while I have no recollection of what threads o conversations I'm involved in. I look up my mentions and work through them. If I don't see a mention, I may miss your post.

The crank clown can't understand what the rest of humanity understands:

AN ordering of the children is not the ONLY ordering of the children.

And back to 'The Adventures Of The Crank Radio Hour':

Crank: Hey boss, I put our sales products in this spreadsheet in the order.

Boss: Which order?

Crank: The order.

Boss: Order by revenue or by items sold or by catalog number or what?

Crank: You know, the order.

Boss: Remind me how you got this job.

It is simple indeed:


x is an element iff Ey xey

x is a class iff (x=0 or Ey yex)

x is a proper class iff (x is a class & ~Ey xey)

x is a set iff (x is a class & Ey xey)

x is an urelement iff x is not a class


Classical set theory theorem: Ax x is a set

Classical class theory theorem: Ax x is a class & Ex x is a set & Ex x is a proper class

Set theory with urelements theorem: Ex x is a set & Ex x is an urelement

Fairy tale characters are an abstract universal. They are general, and they don't actually exist.

Cinderella is a particular fairy tale character. She doesn't exist either, but she is an INSTANCE of the category of fairy tale characters.

Fairy tale characters are abstract universals, and Cinderella is an abstract particular.

In your world you don't have any abstraction at all. I think you're taking a point too far. — fishfry

Nicely said.

I never said any such thing. I've said the opposite — TonesInDeepFreeze

I know you never said such a thing. You mix up physical objects and mathematical objects as if there is no difference between them, and as if the law of identity would apply to both equally. That's why I call you a sophist. It was fishfry's principle, that elements of a set are not physical objects.

Actually I am wrong about that TonesInDeepFreeze showed me the error of my ways. All sets satisfy the law of identity. If I have a set X, I may write X = X by way of the law of identity. I do not need the axiom of extensionality for that. Perfectly clear to me now. — fishfry

No, Tones was referring to the principle called "the identity of indiscernibles", which is completely different from the law of identity. The law of identity makes a thing's identity the thing itself, the identity of indiscernibles associates a thing's identity with the thing's properties. These are fundamentally different principles.

The law of identity applies to sets. So this line of argument is null and void. — fishfry

No, you simply fell for the sophistry. Tones is very good at it, and apt to convince others, earning the title "head sophist".

Yes, very good. A group is any mathematical structure that obeys the axioms for groups. A set is any mathematical object that obeys the axioms for sets. — fishfry

So, a set is a mathematical structure. How do you make this consistent with the head sophist's claim that the members of The Beatles is a set, and the particles which make up a rock is a set? The sophist says "the set whose members are all and only the bandmates in the Beatles has 24 orderings". Notice that this is not stated as possible orderings, it is stated as the "orderings"

Remember your schoolkid example? You recognized that the objects which bear that name have what you called SOME order, and this is an expression of the condition which they are actually in, at any point in time. I would call this their "actual order". Can you see what the head sophist has done? The sophist has removed any distinction of an actual order, to say that the group, or set, has 24 orderings, and all these orderings are equal, or the same, being in each case a different presentation of the same set. But you and I recognize, that in reality there is "SOME order", an actual order, which is the order that the objects are actually in, at any given point in time. The sophist might talk about 24 orderings, but you and I recognize that if these 24 account for all the possibilities, only one of those possibilities represents the very special "actual order", and, that since these elements are physical objects, there must be an actual order which they are in, at any given time.

The law of identity is very important to recognize the actual existence of a thing, and its temporal extension. Through time a thing changes, and the law of noncontradiction stipulates that contradicting properties cannot be attributed to the same thing at the same time. So if a specific group has ordering A at a specified time, that is a property of that group, and it surely cannot have ordering B at the same time. The head sophist claims that the specified group has 24 orderings, all the time (as time is irrelevant in that fantasy land of sophistry). Obviously the head sophist has no respect for the law of noncontradiction, and is just making contradictory statements, in that sophistic fantasy.

That is what happens when we allow that abstractions such as mathematical structures have an identity. Inevitably the law of noncontradiction and/or the law of excluded middle will be violated. Charles Peirce did some excellent work on this subject. It's a difficult read, and you've already expressed a lack of interest in this subject/object distinction, so you probably don't really care. Anyway, here's a passage which begins to state what Peirce was up to.

The relevance of all this to the principles of excluded middle and contradiction is as follows. Peirce wrote that “anything is general in so far as the principle of excluded middle does not apply to it,” e.g., the proposition “Man is mortal,” and that “anything” is indefinite “in so far as the principle of contradiction does not apply to it,” e.g., the proposition “A man whom I could mention seems to be a little conceited” (5.447-8, 1905). If we take Peirce to have meant LEM and LNC, then it appears that he wanted to deny the principle of bivalence (according to which all propositions are true or else false) with regard to universally quantified propositions, and that he meant to claim that existentially quantified propositions are both true and false. But why think that “Man is mortal,” which seems to be straightforwardly true, is neither true nor false? And why think that one and the same proposition, “A man whom I could mention seems to be a little conceited,” is both true and false? Once we see what Peirce meant by “principles of excluded middle and contradiction,” we see that this is not what he was claiming. — Digital companion to C. S. Peirce

http://www.commens.org/encyclopedia/article/lane-robert-principles-excluded-middle-and-contradiction

Sets are subject to the law of identity. — fishfry

This is blatantly untrue, and as demonstrated above, if we assign "identity" to a set, the law of non-contradiction will be violated. The law of identity enables us to understand an object as changing with the passing of time, while still maintaining its identity as the thing which it is. Sets have distinct formulations existing all the time, which would cause a violation of the law of noncontradiction if we allow that a set is subject to the law of identity. Therefore we must conclude that sets are not subject to the law of identity. The type of thing which the law of identity applies to is physical objects. And there is obviously a big difference between physical objects and sets, despite what head sophist claims.

YOU have problems with the empty set. I have no such problems. — fishfry

You also have no problem with contradiction, it seems.

An element is a set in a set theory without urelements. We say x is an element of y if we can legally write x∈y

∈

. Nothing could be simpler. — fishfry

This tells me nothing until you explain precisely what ∈ means. To me, you are simply saying that x is an element of y if x is an element of y. What I am asking is what does it mean "to be an element".

Cinderella is a particular fairy tale character. She doesn't exist either, but she is an INSTANCE of the category of fairy tale characters.

Fairy tale characters are abstract universals, and Cinderella is an abstract particular. — fishfry

If we go with this definition, you ought to se very clearly that sets, as categories, abstract universals, do not have an identity according to the law of identity. A category is not a thing with an identity.

Obviously this does not work. As you said already, elements are often sets. Therefore you cannot characterize the set as an abstract universal, and the element as an abstract particular, because they're both both, and you have no real distinction between universal and particular. There's no point in trying to justify the head sophist's denial of reality. If "Cinderella" refers to a particular, an instance of the category "fairy take characters", then that is a physical object. If "Cinderella" refers to a further abstract category, like in the case of "red is an instance of colour", then it does not refer to a particular. The head sophist seems to have convinced you that you can ignore the difference between a physical object and an abstraction, but you and I both know that would be a mistake.

The sophist crank says, "the principle you stated, the elements of a set are not things".

I never said any such thing. I've said the opposite. — TonesInDeepFreeze

Then the sophist crank says "I know you never said such a thing. You mix up physical objects and mathematical objects as if there is no difference between them, and as if the law of identity would apply to both equally."

That's another LIE from the crank.

That the law of identity applies to both numbers and rocks does not entail that there is no difference between numbers and rocks! It does not entail that there is no difference between abstractions and concretes. The laws of traffic apply to both domestic vehicles and foreign vehicles, but that doesn't entail that there's no difference between domestic vehicles and foreign vehicles! The crank can't reason successfully in even the most basic ways!

Moreover, I did not say that an element of a set cannot be a concrete thing. The set of pencils on my desk has only concrete things as members.

The sophist crank is as usual abysmally confused and making false claims about what I've said.

/

When I first used the term 'sophist crank' I knew I was indulging redundancy', since cranks are by nature sophists. But I've been doing it anyway, to stress the point. It's clear enough by now, though it's been clear enough about him for years.

/

Then the crank, in his usual manner of self-serving sophistry, misconstrues @fishfry. fishfry didn't contradict that the law of identity is different from the identity of indiscernibles.

/

The crank says that the bandmates in the Beatles don't provide for a set. But they do, as they provide for the set {George, Ringo, John, Paul}. The crank can't understand what even a child can understand.

{the pencil on my desk, the pen on my desk} is a set whose members are of concretes and it has two orderings.

{1, 2} is a set whose members are mathematical objects and it has two orderings.

And even if we demurred from saying that such things as number are abstract objects, then still the principle that there is more than one ordering of a set obtains, since we may adduce sets whose members are concrete objects.

So, what example would the crank give of a set with more than one member? Whatever example the sophist gives, that set has more than one ordering.

Then the crank says "a set is a mathematical structure". That is an example of arguing by mere insistence on one's personal definition. Typical sophistry. Of course, one may stipulate any definition one wants to stipulate. But that carries no argumentative import in context of use of the word with a different definition. In mathematics and even in everyday life, the word 'set' is not ordinarily used to mean 'a structure'. However, mathematics does also address the notion of structure, and provides rigorous definition; but the crank, in his obdurate willful ignorance knows nothing about that, as he knows nothing about the mathematics he incessantly gets completely wrong.

Then the crank points out that I said the set has 24 orderings and that I did not say it has 24 possible orderings. That is, typically, an inane objection by the crank. (1) Extensional mathematics does not use intensional modalities. (But there are systems of intensional mathematics too.) (2) Even if we do speak instead of 'possible orderings', any particular one of those possibilities is not the only possibility, so it is still not THE ordering. It is merely one of the "possible" orderings chosen for our consideration. Any other "possible" ordering could be chosen and then, following the crank's notion, it would have to be considered to be THE ordering. So there would a different THE ordering depending on which ordering we happen to choose for consideration, which is still incoherent.

So, even deferring to the crank's insistence about "possible", which of the 24 possible orderings of the set whose members are the bandmates in the Beatles is THE ordering of that set?

I've given the crank the following information about half a dozen times already, but like the horse led to water who will not drink, the crank will not think (apologies to D. Parker):

Yes, we can specify a particular ordering of a set and refer to that set vis-a-vis that specified ordering. For example, let B be the set whose members are all and only the bandmates in the Beatles, and let R be the ordering of B alphabetically by first name. Then we have the STRUCTURE <B R>. That accords with the notion of a set along with a particular ordering.

The crank says, "there is "SOME order", an actual order, which is the order that the objects are actually in, at any given point in time.

At this exact moment of time, there are two orderings of the set of writing tools on my desk:

{<pencil pen>}

and

{<pen pencil>}

At this exact moment of time, there are two orderings of the two kids on the playground:

{<Joe Maya>}

and

{<Maya Joe>}

/

Then the crank goes on with yet more confusions. Reading his posts, I am reminded of a character in 'The Office' saying about the on and on, full of it fool, Michael Scott, "Where's the off button on this moron?"

This deserves to be especially highlighted:

The crank says, "[Tones In Deep Freeze] has removed any distinction of an actual order, to say that the group, or set, has 24 orderings, and all these orderings are equal, or the same"

That is yet another flat LIE from the crank, and a really stupid lie.

I never said that 24 orderings are the same or that they are equal. That would be a ridiculous thing to say. Indeed it is my point that they are not the same. There are 24 different orderings. Of course they are not all the same orderings. The crank is so mentally inept that he can't distinguish between (1) there are 24 different orderings that each have the property of being an ordering of a certain set and (2) all those orderings are the same.

The cranks daily posts rank garbage on a ... PHILOSOPHY forum!

Moreover, I did not say that an element of a set cannot be a concrete thing. The set of pencils on my desk has only concrete things as members. — TonesInDeepFreeze

I know, that's the problem. For you, a set may consist of concrete things, or it may consist of abstractions, because in your sophistry you do not differentiate between the two. Then you claim that there is no order to the concrete things which compose a set, when in reality there is.

I never said that 24 orderings are the same or that they are equal. That would be a ridiculous thing to say. Indeed it is my point that they are not the same. There are 24 different orderings. Of course they are not all the same orderings. The crank is so mentally inept that he can't distinguish between (1) there are 24 different orderings that each have the property of being an ordering of a certain set and (2) all those orderings are the same. — TonesInDeepFreeze

Right, continue in your violation of the law of noncontradiction. The same set has contradicting properties, i.e. different orderings. Good one bro, I hope that's just your sense of humour again.

The crank says, "For you, a set may consist of concrete things, or it may consist of abstractions, because in your sophistry you do not differentiate between the two."

Stop lying, crank.

So which is it then? Does a set consist of concrete things, or does it consist of abstractions?

I never said that 24 orderings are the same or that they are equal. That would be a ridiculous thing to say. Indeed it is my point that they are not the same. There are 24 different orderings. Of course they are not all the same orderings. The crank is so mentally inept that he can't distinguish between (1) there are 24 different orderings that each have the property of being an ordering of a certain set and (2) all those orderings are the same. — TonesInDeepFreeze

The crank mindlessly replies "Right, continue in your violation of the law of noncontradiction."

The crank is so mentally deficient that he can't see that it's not a contradiction that "there are 24 orderings of a set" does not imply "all those orderings are the same". It's an incorrect implication, because, indeed it is a contradiction to say that 24 different orderings are all the same ordering. It seems it is in the crank's imagination that, somehow 'different' implies 'same'. That's his problem, not mine, since indeed, for me 'different' does not mean 'the same'.

The crank's illogic and utter obtuseness are not less than stunning. And in a philosophy forum!

Now like a child with an attention disorder, the crank asks me whether the members of a set are abstractions or concretes, after I explicitly said that they can be either, and I gave explicit examples. Is the crank not able to read?

The crank is so mentally deficient that he can't see that it's not a contradiction that "there are 24 orderings of a set" does not imply "all those orderings are the same". — TonesInDeepFreeze

The orderings are different, and contradictory properties of the set. And, it is a violation of the law of noncontradiction for that set to have those contradictory orderings.

Now like a child with an attention disorder, the crank asks me whether the members of a set are abstractions or concretes, after I explicitly said that they can be either, and I gave explicit examples. Is the crank not able to read? — TonesInDeepFreeze

Exactly as I said, you fail to provide a differentiation between concrete objects and abstractions. Why did you say I lied about this?

If a set consists of concrete objects, then it has the order that those concrete objects have, and no other order. To say that the set has other orderings is to mix up concrete objects with abstract objects in the way of sophistry.

It is not any more a contradiction for a set to have more than one ordering than it is a contradiction for a person to own more than one hat.

It's not a matter of whether I explicate the difference between concrete and abstract. Rather, whatever one's explication of the difference, abstract objects can be elements and concrete objects can be elements.

The crank argues by persistently ignoring the rebuttals, examples and explanations given him:

The crank says, "If a set consists of concrete objects, then it has the order that those concrete objects have, and no other order."

That was answered posts ago by me. The crank can't or won't read the posts he replies to.

The crank asks, "Why did you say I lied about this?"

Here are the lies:

"You mix up physical objects and mathematical objects as if there is no difference between them"

I explicitly mentioned that a number is abstract and a rock is concrete. And I reiterated that. The crank is a liar.

"For you, a set may consist of concrete things, or it may consist of abstractions, because in your sophistry you do not differentiate between the two." [the lie bolded]

I have never conflated abstractions with concrete. The crank is a foolish liar.

"[Tones In Deep Freeze] has removed any distinction of an actual order, to say that the group, or set, has 24 orderings, and all these orderings are equal, or the same"

I do not at all make any such utterly ridiculous statement that the orderings are the same. And I reiterated that. The crank is a profoundly illogical liar.

The crank enacts one of the starkest examples of mentally pathological illogic I've seen in a while:

I say, in clear, emphatic, and unequivocal terms that the 24 orderings are different orderings. It is at the heart of my point that they are different orderings. It would be absurd to say that they are not different orderings. But the crank says that I say that they are the same ordering.

Illogic doesn't get much more dire than the crank's.