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.