Another common crank fallacy is claiming that mathematics is false by way or arguing that mathematics uses words in ways different from their ordinary meanings or different from their meanings in certain other fields of study.

By the way, the distinction between the countability of the naturals and the uncountability of the reals doesn't, in a certain important sense, even require infinitistic assumptions.

Even if there is no set of all the natural numbers and no set of all the real numbers (and even if real numbers are not infinitistically conceived as equivalence classes of Cauchy sequences or Dedekind cuts), still we may note that there is no procedure such that its outputs are increasingly longer finite sequences of real numbers (however real numbers are finitistically conceived or represented such as a real number itself as a procedure for outputting increasingly longer finite decimal sequences) and such that, for any real number, it will eventually be an output. (I think that's right, but perhaps there could be objections?)

Regarding the fact that mathematics is not even isomorphic to a system of physical objects:

It's not intended to be, and it's not required for the efficacy of mathematics. Detailed explanations of that point have been given over the years in this forum.

If one rejects the view that abstract objects exist (and obviously, as abstractions, they don't exist physically), then, of course, the left term and the right term in an identity statement cannot refer to abstract objects. But that is a different objection than objecting to taking '=' as standing for the identity relation.

And if one objects to calling whatever mathematics refers to as 'objects', then we note that the word 'object' is a convenience but not necessary, as we could say 'thing' instead, or 'value of the term', or 'denotation of the term', or even none of that, and just say 'members of the domain of discourse' so that 'T = S' is interpreted as, for any model M for the language, M(T) is M(S).

It is not required to have any particular ontological view of what mathematical terms refer to just to understand that '=' stands for the identity relation. That is, whatever the terms T and S refer to (no matter what one regards mathematical terms as referring to), we understand that 'T = S' stands for the statement that whatever 'T' stands for is the same as what 'S' stands for.

Moreover, there is a difference between what is meant in mathematics by '=' and what one thinks mathematics should mean by '='. Whatever one thinks mathematics should mean by '=' doesn't change the fact that in mathematics '=' stands for identity.

You may try for, literally, years and he will not understand.

n — Mark Nyquist

I thought you meant an intermediate step in the proof. I have no comment on your characterization of phases.

The problem is not misnamed. I explained why it is a paradox from certain assumptions or axioms. Also, to be clear, 'paradox' is not a technical mathematical term. It's a general term for the situations I mentioned: From supposedly acceptable assumptions, principles or axioms, we derive a contradiction or even just a highly counter-intuitive implication.

I gave the Mark Twain / Samuel Clemens example as an illustration, not an argument, of the distinction between sense and denotation. And I mentioned the distinction between sense and denotation not as an argument for the point that, in mathematics, '=' stands for identity, but rather to refer to another aspect of the matter. And this point is not confined to physical objects.

The value represented by A is identical to the value represented by B. — Michael

A = B

A is B.

The value named by 'A' is the value named by 'B'.

A is equal to B.

The value named by 'A' is equal to the value named by 'B'.

A is identical to B.

The value named by 'A' is identical to the value named by 'B'.

Seven ways of saying the same thing.

But you will never bring the crank to understand that.

Most of your own posts are filled with distortions. — Corvus

You argue by mere assertion.

Anyway, you said this in not worth your time and signed off with "All the best", yet you're still going at it.

I am only replying to your posts, the way they are. — Corvus

Whatever that might mean in your own mind.

Laurel and Hardy — Corvus

As long as I can be Laurel. Stan Laurel is a great hero of mine. Right up there with Buster Keaton.

Stop distorting the facts — Corvus

You've not shown that I've distorted any fact. Meanwhile, you've been distorting all over the place, as I have shown.

It sounds like you are a little string controlled doll in Banno's pocket. — Corvus

First you say I speak for Banno, then you say that Banno controls me. But if Banno controls me, and I speak for him, then I speak for him at his control, so then it should be just fine for me to speak for him. (Though I don't speak for him and he doesn't control me.)

You're lying again. I committed no action that constitutes speaking for Banno.

You do. But of course you won't admit it. — Corvus

I haven't presumed to speak for Banno.

You're lying again.

You speak for Banno — Corvus

I explicitly said I do not speak for Banno.

You say that in mathematics 'infinite' means 'finite', but 'infinite' means 'not finite'. Then I say that I do not speak for Banno and you say that I do. I think the problem might be that you don't know what the word 'not' means.

and now trying to speak for me — Corvus

I haven't presumed to speak for you.

Meanwhile, you've put words in my mouth, and failed to recognize that when I caught you doing it.

It seems obvious your whole purpose of coming into the forum is forcing people to admit errors when the error is on your side. — Corvus

That's a stupid thing to say.

So sorry you got mixed up about my view of the existence of the Russell set. — Mark Nyquist

I didn't get mixed up. You were mixed up.

I was developing an alternative method using the concept of mathematical objects as proposed, existent or non-existent. — Mark Nyquist

I had no comment on that. Rather, your replies regarding the mechanics of the ordinary proof were confused.

Would it be fair to say your view develops the Russell set as a proposed mathematical object and concludes that it is ultimately a non-existent mathematical object? — Mark Nyquist

No. The way you say it is kind of along the lines of what I say, but I don't bring along notions of "proposed object" and "ultimately non-existent".

Rather, as I explained, it is a proof by contradiction, keeping it precise and simple, without the extraneous "proposed object" and "ultimately non-existent":

Toward a contradiction, suppose there is an x such that for all y, y is a member of x if and only if y is not a member of y.

Then x is a member of x, and x is not a member of x.

Therefore there is no x such that for all y, y is a member of x if and only if y is not a member of y.

I am relying on your intermediate conclusion that the Russell set does not exist to go straight to the final conclusion that if the Russell set does not exist — Mark Nyquist

What? What you call my "intermediate conclusion" is not intermediate, and it is the same as what you call the "final conclusion".

The theorem ("the final conclusion") proved is:

There is no x such that for all y, y is a member of x if and only if y is not a member of y.

Put more informally, "There is no set whose members are all and only those sets that are not members of themselves".

Put another way, the name "The Russell set", which is supposed to name a set whose members are all and only those sets that are not members of themselves, does not properly refer to anything.

a paradox does not exist — Mark Nyquist

The paradox was that, from the ordinary view (once given as an axiom) that for every property there is the set of those things having that property, we derive a contradiction.

When we eschew that view (and the axiom that captures it) that for every property there is the set of things having that property, we are not burdened with the contradiction that that view (and axiom) entails.

Summary,
The Russell set does not exist.
Based on the proposed defined mathematical object failing by contradiction. — Mark Nyquist

That's not how I would say it, but it's close enough. I won't split hairs to quibble with it.

In defining the Russell set, two or more (known to exist) defined mathematical objects are used to define the Russell set — Mark Nyquist

I don't know what you're referring to.

The definition is:

R = {y | y is not a member of y}

i.e.

R is the set of all and only those y such that y is not a member of y.

But it's an improper definition, because there is no set whose members are all and only those y such that y is not a member of y.

How can one admit error when he is not in error but the other party is? — Corvus

Even if the other party were in error (which is not the case here anyway), if you are also in error, then you could admit it.

Actually, it seems you can't.

You compound your errors now by claiming that you've not been in error, when its overwhelmingly clear that you have been, and in so many ways.

You [Banno] haven't even explained what "infinity" means. — Corvus

I don't speak for Banno, but I have said that there is no set named with the noun 'infinity', but rather there is the adjective 'is infinite' defined:

x is infinite iff x is not finite

"Incorrigible" would be more accurate. — Banno

Amazing in the forms of incorrigibility.

If you thought he had little to do with math — Corvus

You're lying about me. (Though you weasel with 'if'.)

I never said that Wittgenstein has little to do with mathematics.

It's overwhelmingly the case that Wittgenstein is one of the very most central philosophers in the subject of logic and mathematics.

That I say he does not speak for mathematics in the sense that mathematics speaks for itself in its definitions, I am not at all saying that it is not relevant to mention what he says about mathematics. He has things to say about mathematics, but he cannot be fairly regarded as speaking for mathematics, especially as the fact that he was critical of the notion of infinity in mathematics does not imply that mathematics regards 'infinite' to mean 'finite'.

You don't seem to even know what said what, and what was whose points — Corvus

You have not shown that I've failed to know what has been said.

and just get into ad hominem all the time. — Corvus

When you say "just", you're lying.

I post extensive arguments that are not ad hominem, and give extensive information and explanation that is not ad hominem.

Anyway, I do comment on the ignorance, confusion and dishonesty of cranks, but not as an ad hominem argument in the sense of something like "you are [fill in the personal remarks here] therefore your argument is not sound." Rather, I have given you fulsome information, explanation, counter-arguments and refutations, no matter what else I have to say about your ignorance, confusion and dishonesty. Pointing out that you are a crank is merely lagniappe to my substantive comments.

Would you say your postings are high standard? — Corvus

They're never good enough for me, for a number of reasons. But they do provide a lot of information, explanation, and cogent arguments, and sometimes have other virtues too. Posting is hard, because it's impromptu and usually under the constraint of being time diverted from other things one wants and needs to do. So, that considered, all and all, I think I've written some really good posts, while others are just okay, but in just about all cases, I wish that I had time to make them a lot better still. And too many typos in them.

You don't even know what Wittgenstein was up to. — Corvus

You don't know what I know about Wittgenstein, including what I've forgotten and would need to refresh myself on. And it's aside the point anyway, as my arguments are not about Wittgenstein, not even to try to dissuade you from sharing his views about mathematics. Whatever the case about Wittgenstein, it is not the case that mathematics regards 'infinite' as meaning 'finite', for example.

it was clear that W. had rejected the concept of infinity
— Corvus

That doesn't entail that in mathematics 'infinite' means 'finite'. What in all creation is wrong with you?
— TonesInDeepFreeze
You better ask Wittgenstein what he meant by that. I have my own point. What with you? — Corvus

You are amazing!

You said that mathematics takes 'infinite' to mean 'finite'. Then you said that actually you meant that Wittgenstein said that. Then when you are offered that you can provide any context or explanation to support that, you say that I should I should ask Wittgenstein rather than indeed, you providing an argument that when Wittgenstein said that discussions about infinity are finite, he implied that mathematics takes 'infinite' to mean 'finite'. And then you totally reversed to say as much as that whatever Wittgenstein's point is, you have your own point. So which of these things that you've already said not is that you mean?:

(1) Mathematics regards 'infinite' as meaning 'finite'.

(2) Wittgenstein implies that mathematics regards 'infinite' as meaning 'finite' and you agree with that.

(3) Maybe Wittgenstein did not imply that mathematics regards 'infinite as meaning 'finite', but that you do claim that mathematics regards 'infinite' as meaning 'finite'.

And it's not been at issue that Wittgenstein was critical of the notion of infinity in mathematics.

So what? It doesn't say that mathematics takes 'infinite' to mean 'finite'. And even if it did (which it does not), it doesn't represent mathematics or mathematicians, since they very certainly do NOT take 'infinite' to mean 'finite'.
— TonesInDeepFreeze
It was just to let you know it was what Wittgenstein was saying, and he was a great philosopher of language, logic and mathematics. — Corvus

First, you said that mathematics takes 'infinite' to mean 'finite'. You didn't say anything about Wittgenstein there. Then, you said that you expected me to infer that you meant Wittgenstein, though there was no hint even about him there. Then, you mentioned a post in another thread where you quoted Wittgenstein commenting that discussions about infinity are finite. While, it may be that he meant that as part of his argument against the notion of infinity, at least at face value, it is not an assertion that mathematics regards 'infinite' to mean 'finite'. Finally, even if that was what he meant, he does not speak for mathematics, which speaks for itself when it defined 'infinite' as 'not finite, and not ludicrously as 'finite'.

And then your quoting of Chat GPT as part of your bizarrely specious attempt to dispute my explanation of how mathematics actually does define 'infinite'.

And your really foolish dispute against the fact that modern, current, authoritative, standard and widely referenced textbooks in the main areas of mathematics define 'infinite' as 'not finite'.

And that's all just recent posting by you, not mentioning all the other garbage you've posted in this thread and at least another.

You don't seem to know anything about Wittgenstein anyway from your posts. — Corvus

I haven't made any claims about him, other than that, at least at face value, "discussions are finite" does not mean that mathematics regards 'infinite' as meaning 'finite'.

He was also a student of Russell too, and both were deeply into mathematics and logic. — Corvus

So what?

As I said, Wittgenstein does not speak for mathematics. Mathematics speaks for itself when it defines 'infinite' as 'not finite' and not, ridiculously 'finite'.

you just keep on writing disinformation in your posts without even checking it — Corvus

You've not shown any disinformation in what I posted.

it was clear that W. had rejected the concept of infinity — Corvus

That doesn't entail that in mathematics 'infinite' means 'finite'. What in all creation is wrong with you?

Infinite in mathematics means "finite". Hence their discussion will end. — Corvus

You are claiming again that in mathematics 'infinite' means 'finite'.

You have a bizarre idea that because Wittgenstein was critical of the notion of infinity in mathematics that therefore mathematics takes 'infinite' to mean 'finite'. Amazing.

It is not just mathematician's discussions which end. All discussions end. That is too obvious.

What Wittgenstein must have meant was the concept of infinity in mathematics. — Corvus

That might be the case. That might be part of Wittgenstein's argument against the notion of infinity. I don't know. But even if it is, it still is not saying, at least at face value, that mathematics regards 'infinite' to mean 'finite'.

So which discussion is not finite in that case? — Corvus

What? Are you trolling?

Banno didn't say that discussions are not finite. He is saying that "discussions are finite" doesn't mean that mathematics takes 'infinite' to mean finite'.

"Let us not forget: mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end." - Philosophical grammar, p483. Wittgenstein.
— Corvus
Wasn't he saying clearly mathematician's infinite are finite? — Corvus

If he's saying that there, then he's definitely not "clearly" saying it. I don't claim to know what he is driving at. But at least at face value, the sentence does not read to be saying that in mathematics 'infinite' is taken to mean 'finite'. But you're free to provide more context and analysis to justify your interpretation.

It was Frege, Russell, Quine who had reservations on it even if didn't oppose to it. — Corvus

I addressed that. You SKIPPED it.

If you have something to say specific about those mathematicians/philsophers, then please say what it is.

I asked for the textbook definition for infinite in math. — Corvus

And I gave it to you! In detail. With clear, exact explanation.

Again, if I list for you the titles and authors of the many textbooks that are currently standard, even quintessential references in the subject, will you finally admit that you are incorrect in the ridiculous claim that in mathematics 'infinite' means 'finite?

You misunderstood. It meant that Wittgenstein said that mathematician's infinite means finite in his writings. — Corvus

You said that mathematics regards 'infinite' to mean 'not finite'. You didn't say anything about Wittgenstein there. If by saying that mathematics takes 'infinite' to mean 'not finite' you actually mean something different, such as that Wittgenstein notes that mathematical discussions are finite, then you need to write that and not that mathematics takes 'infinite' to mean 'not finite' and not to then blame readers for your error.

Moreover, I don't opine on what Wittgenstein meant in that quote of him, but at least, at face value, saying that discussions are finite is not the same as saying that mathematicians mean 'finite' when they write 'infinite'.

Tone was in the thread, and he would have seen it. — Corvus

So what? It doesn't say that mathematics takes 'infinite' to mean 'finite'. And even if it did (which it does not), it doesn't represent mathematics or mathematicians, since they very certainly do NOT take 'infinite' to mean 'finite'.

The quote below deserves attention as among worst:

I would have expected your reply to my question from the reputable and well known modern math textbooks which says "infinite" is "not finite" — Corvus

The textbooks I can cite you are not just reputable, but they are among the most standard, most used, and most referenced textbooks in current use.

I asked you why you want me to name one if you are not interested in looking at it. Indeed, not interested in looking at any of the many I can cite.

On my desk right now, I have a stack of modern textbooks, some of them regarded as quintessential references, in various mathematical subjects, as they all define 'infinite' as 'not finite'. What do I get if I list their titles and authors for you? You're not going to look them up. So what's the point? Or, how about this: I'll list them all, then you can admit that that you don't know what you're talking about when you say that mathematics defines 'infinite' as 'finite' but rather that mathematics defines 'infinite' as 'not finite'.

I am not claiming anything on the math theory. — Corvus

You're lying in our face. You claimed that in math 'infinite' means 'finite'.

I am just pointing out the contradictions and false information in your posts, and replying to them. — Corvus

You have shown no contradiction or false information in my posts. Rather, you have shown that you are ignorant of the subject, confused about the subject, disinformational in things you've said about the subject, specious in your arguments about the subject, willing to resort to ridiculously intellectually incompetent posturing by quoting Chat GPT (!) that you don't understand and as the quotes don't even approach impeaching anything I've said, and flat out lying when you say that you didn't make a claim about mathematics when plainly you did.

It would be a gross distortion of the fact and over exaggeration to state anything more than that about my replies. — Corvus

Nope. If anything it's understatement to say what I've said about your intellectual incompetence and dishonesty.

I have already quoted from Wittgenstein from his writings "infinite" in math means "finite" — Corvus

(1) Please link to where you quoted Wittgenstein writing that 'infinite' in mathematics means 'finite'.

(2) Wittgenstein doesn't speak for mathematics anyway. Whatever Wittgenstein wrote, it wouldn't change that fact that mathematics does not define 'infinite' as 'finite', which would be utterly ridiculous, as mathematics defines 'infinite' as 'not finite'.

But my point is not about "infinite" is "finite" or whatever. — Corvus

Whatever your point is, what you claimed that in mathematics, 'infinite' means 'finite', which is a wildly ridiculous claim and blatant disinformation.

My point was that the concept "infinite" means something totally different, and math's infinity in set theory doesn't exist. — Corvus

That's a different claim from the claim that, in mathematics, 'infinite' means 'finite'.

But you are quoting from the old and outdated mathematician Dedekind on the concept of "infinity", and it means "not finite". — Corvus

You are hopeless as far as rational discussion.

You asked me what textbooks in mathematics define 'infinite' as 'not finite'. The answer to that question is that just about every textbook in mathematics that gives a mathematical definition of 'infinite' gives the definiens as 'not finite', or sometimes the Dedekind definition that is equivalent with 'not finite' in mathematics. The formal mathematical definition 'not finite' goes back to Tarski and the definition 'one-to-one with a proper subset of itself' goes back to Dedekind; but that in no way vitiates that still, the current definition is 'not finite' or its Dedekind equivalent. You challenged me as to what mathematics textbooks say; my answer to that is not vitiated by the fact that the definition is long standing in mathematics. What is wrong with you?

Moreover, obviously, in even just an ordinary context, 'the 'in' in 'infinite' is a negation, so it's 'not finite' and not 'finite'. Again, what is wrong with you?

And now you are pasting Chat GPT (!) quotes that you don't even understand. You can't learn set theory from Chat GPT! What is wrong with you?

"Dedekind's set theory lacks a formal axiomatic foundation comparable to other set theories like Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). Without a clear set of axioms, Dedekind's set theory may be seen as less rigorous or formal by contemporary standards." - Chat GPT — Corvus

You are such an intellectual incompetent.

I am not talking about Dedekind's theory. I'm talking about a particular definition. And that definition is used is equivalent with 'not finite' in the ZFC that you just mentioned. Indeed, any informal theory is less rigorous than formalized ZFC. So what? That doesn't change the fact that Dedekind's formulations cannot or have not been formalized subsequent to his own writings.

Again, since you missed this:

Definition of 'infinite' in mathematics:

x is finite iff x is one-to-one with a natural number.

x is infinite iff x is not finite (by the way, that is sometimes called 'Tarski's definition)

Another definition of 'infinite' in mathematics:

x is infinite iff x is one-to-one with a proper subset of x (Dedekind's definition)

Those are provably equivalent in set theory with the axiom of choice (such as ZFC). Without the axiom of choice, we can only prove: If x is one-to-one with a proper subset of itself then x is not one-to-one with a natural number.

In any case, with both those definitions, it is blatant that 'x is infinite' is defined as 'x is finite'.

Moreover, now you are taking recourse to the notion of formalization, when just a few posts ago you were trying to dispute me when I mentioned a key advantage of formalization! What is wrong with you?

"While Dedekind made significant contributions to the understanding of infinity, his treatment of infinity in set theory may be considered less systematic compared to later developments, such as Cantor's work on transfinite numbers and ZFC set theory." - Chat GPT — Corvus

Cantor was more systematic about sets, but Cantor also was not a formal theory and had problems that needed to be rigorously resolved by formal set theory.

Anyway, this has no bearing on the fact that the set theoretical definition of 'infinite' is not ridiculously, as you claim, 'finite', nor on the fact that both Tarski's and Dedekind's definition obtain in current mathematics.

"Some critics argue that Dedekind's definition of infinite sets as those that can be put into one-to-one correspondence with proper subsets of themselves is not as precise or comprehensive as later formulations." - Chat GPT — Corvus

Ask Chat GPT who it thinks those critics are and to quote them.

Dedekind's definition incorporated into axiomatic set theory is absolutely precise. And in ZFC it is exactly as comprehensive as Tarski's definition, since in ZFC they are equivalent, as I stated that equivalence explicitly in my previous post.

You don't know jack aboutany of this. You just want to be right about disdain for set theory, so you're willing to enter any specious and counterfactual argument you can come up with, including inapposite quotes from.. Chat GPT (!).

"Dedekind's set theory does not provide a set of explicit axioms like those found in ZFC set theory. This lack of a formal axiomatization can make it difficult to establish the foundational principles of Dedekind's theory and to reason rigorously about sets within this framework." - Chat GPT — Corvus

Again, that does not vitiate that nevertheless his work has been formalized subsequent to his own writings and that include his definition of 'is infinite'.

You are foolishly quoting Chat GPT without even a basis to understand the quotes, their context or their import or lack thereof for our conversation.

"While his work laid important groundwork for the development of modern set theory, it may not encompass the full range of concepts and techniques found in more contemporary approaches." — Corvus

So what? Contemporary mathematics still uses his definition of 'is infinite' as it is equivalent with 'not one-to-one with a natural number', i.e. 'not finite', in ZFC.

Which math textbook says "infinite" means "not finite"? — Corvus

You said, "Problem with Set Theory is that their concept "infinite" means "finite""

What set theory textbook, or any reference in set theory or mathematics, says that 'infinite' means 'finite'?

Meanwhile, many textbooks in mathematics, including set theory, analysis, algebra, topology, computability, probability and discrete mathematics give the definition of 'infinite' as 'not finite'. What is your purpose in asking if you're not thinking of reading one of them?

Two pairs of definitions:

x is finite iff x is one-to-one with a natural number
and
x is infinite iff x is not finite

x is Dedekind finite iff x is not one-to-one with a proper subset of x
and
x is Dedekind infinite iff x is not Dedekind finite

But with the axiom of choice (as with the most common set theory in mathematics, which is ZFC) we have:

x is finite iff x is Dedekind finite
thus
x is infinite iff x is Dedekind infinite.

More specifically:

Without the axiom of choice, we have:

If x is finite then x is Dedekind finite
thus
If x is Dedekind infinite then x is infinite

With the axiom of choice, we have both:

If x is finite then x is Dedekind finite
thus
If x is Dedekind infinite then x is infinite

and

If x is Dedekind finite then x is finite
thus
If x is infinite then x is Dedekind infinite

/

Meanwhile you are posting flat our disinformation when you post ""Problem with Set Theory is that their concept "infinite" means "finite"".

And you'll not show any textbook or article or lecture notes in set theory or mathematics that say 'infinite' means 'finite'.

Indeed, even in plain language, the prefix 'in' with 'infinite' is taken in the sense of 'not'. x is infinite if and only if x is not finite.

You need to stop posting confusions and disinformation.

Funny thing is that, though I might be mistaken, I suspect that there is an error in the article (though it is not material).

The article says that intuitionistically:

~ReR -> ReR implies ReR.

But I don't see how that is so.

Yes, ~ReR -> ReR implies ~~ReR. But we can't infer ReR from ~~ReR.

So, the only way I could think of doing an intutionistic proof is not by way of:

ReR & ~ReR

but rather by way of:

~ReR & ~~ReR

It might be jolly for you to say what you think is the very first incorrect sentence in the Stanford article.

Just enough to understand the problem. — Philosopher19

Just enough to think you understand it. But more than enough for you to completely mangle it.

I see no point in continuing this discussion. — Philosopher19

You've said that about twenty times already, yet you continue. But you are right. If you are unwilling to carefully read toward understanding, not just skim for an opportunity to misrepresent, an article such as the one at the Stanford Encyclopedia then there is no point for you to be posting about it.

But we are trying to dispell the contradiction, not prove it.

If the Russell set doesn't exist there is no contradiction. — Mark Nyquist

Again, you're mixed up, and likely unfamiliar with proof by contradiction.

We prove that the assumption "there is a set whose members are all and only those sets that are not members of themselves" implies a contradiction thus that that assumption is false, which is to say that there does NOT exist such a set.

If you continue to insist that we're saying that such a set does exist, after three posters have already explained, in detail and in different ways, your misunderstanding, then I'm guessing you're trolling.

EDIT: I see now that after the quoted post, you replied further that you recognize that you don't understand. So, fair enough.

You accept that than the Russel set exists and is legitimate. I don't think it has a sound basis. It's based on definition and that's not proof of existence. You have a burden of proof. — Mark Nyquist

Same as above. You're mixed up and have it reversed.

We very clearly do NOT think there is such a set. Rather, we PROVE that there is no such set.

↪TonesInDeepFreeze
You suggest the Russell set exists only based on the process of defining it. — Mark Nyquist

You are so very mixed up that you are getting this all completely reversed.

No, I do not at all suggest that such a set exists. Rather, we are giving you proofs that such a set does NOT exist.