I see. I thought ZF was ONLY an attempt to bypass Russell’s paradox. I see there’s more to it than that. Thank you for informing me. But I do think Russell’s paradox is a misunderstanding of semantics. Specifically the semantic of ALL:

Take x to be the set of all sets. Take y to be the set of all sets that are members of themselves. Take z to be the set of all sets that are not members of themselves.

x necessarily encompasses ALL sets that are members of themselves (x) as well as ALL sets that are not members of themselves (members of x that are not x itself).

Is z a member of itself? There are two possible answers: Yes, and no:

No: If z is a set, then it is definitely a member of the set of all sets and not a member of itself (this leads to a contradiction because z does not contain itself, so it is not the set of ALL sets that are not members of themselves…precisely because it doesn’t include itself (a set that is not a member of itself)).

Yes: There is only 1 set encompassing all sets that are not members of themselves. x is that set. Just because z is contradictory, doesn’t mean x has to be contradictory too.

Just as PA does not encompass/exhaust ALL numbers, z does not encompass/exhaust ALL sets that are not members of themselves.

We cannot say there is no set of all triangles when triangle is a meaningful term. So if we cannot say there is a set of all sets, we are in fact saying set is not a meaningful term. We cannot afford this move. We cannot have ANY set independently of x. To reject x is to reject ALL sets. Yet, this is what has been mainstream since the 1900s. I believe it is rooted in a misunderstand of ALL and Infinity.

2 consists of 1 + 1
3 consists of 1 + 1 + 1
Trying to count to Infinity consists of 1 + 1 + 1…
Infinity consists of …1 + 1 + 1…

2 + 4 + 6… is greater than 1 + 1 + 1… This is because whilst both finites sets are increasing in size endlessly, one is doing so at a greater rate than the other. The cardinality of neither is Infinity.
…2 + 4 + 6… is exactly the same as …1 + 1 + 1… This is because the cardinality of both is Infinity

All of any increasing is implied to have already occurred. It represents a final state, not an active progression. It is what was already there - not the effort to obtain it.

That is the first step to seeing the final proof concerning this whole issue.

More than two. There’s measure, which lets us rigorously show that the interval [0,1/2] has exactly half the size of the interval [0,1], even though both intervals contain uncountably many points. Another is asymptotic density, in which the evens are exactly half of all the naturals.

It’s no different than size referring to a person’s weight, versus size referring to their height. By one meaning two people could have the same size by weight by not by height. “Size” is ambiguous and must always be qualified. Especially in math, where “size” has no meaning at all except in a given context, which may differ from one context to another.

Ok. Then progress has been made.

What I’m presenting is the standard mathematical point of view as I had beaten into me (/s) by professors at some of the finest universities in the land.

The notion of size in terms of proper subsets is perfectly standard.

Yes but YOU are the one using “size” ambiguously. When mathematicians use the words size, it’s always in a particular context with a particular meaning: proper subset, cardinality, asymptotic density, measure, etc.

I have no problem with that. The proper subset notion of size is certainly useful in some contexts.

Yes. First, the symbol (\infty) is not defined in set theory. It’s only defined in the context of the extended real numbers, and its only use is as a shorthand in limit expressions and so forth. For example so that we can say “the limit of 1/x as x goes to infinity is 0.” Otherwise we’d have to say “the limit as x increases without bound.” It’s also used in measure theory so that we can assign the measure infinity to sets like the entire real line. But the way you are using it, it’s not defined at all.

Secondly, the exponent notation is only defined for finite exponents like (10^4). It’s not defined for infinite exponents.

That’s a divergent infinite series whose value, if you were to define it, would be infinity in the extended real numbers. It would be trivial and of little value. Certainly not the great emphasis that you want to put on it.

This is your personal definition, but it’s not the mathematical definition. What does “number” mean in this case? There certainly are transfinite cardinals and ordinals greater than any integer, but you are not referring to those.

Ok, if you want to use it to represent an arbitrary cardinal or ordinal. But I suspect you don’t mean it that way.

If it makes you happy to say that I won’t argue any more. It makes no mathematical sense but it does have the value of making you happy, and I would never take that away from you. My only purpose here is to clarify some mathematical misunderstandings. But if that’s not your aim, and your aim is to have your own private terminology, I will stop arguing with you about it.

But let me be at least a little bit conciliatory here. If you want to say that (10^\infty = \infty), I can live with that. It just isn’t particularly interesting or useful.

I suppose it depends on whether you want to have your own private terminology, or whether you want to make contact with math as understood by mathematicians. It’s ok either way.

I will make a distinction between two different things. Tell me if you think this to be a meaningful distinction: ‘…’ implies continuing forever (active progression) ‘…’ implies a complete state, not an active progression. So:

A) 1234…
B) 1234…

Do we agree that there is a meaningful difference between A and B?

Consider the following:

I = 1
II = 2
III… = continuously adding one (active progression without end. This will never reach Infinity in quantity)
III… = Infinity in quantity (complete state. Not continuously adding one. Not an active progression without end. Without this (Infinity), III… would not be possible at all)

A*) 1234… versus 2468…

B*) 1234… versus 2468…

I’m confident that we are in agreement that 1234… is exactly half the size of 2468…
Nothing in A* amounts to the quantity of Infinity. Do we agree on this?

123… = IIIIII…
246… = IIIIIIIIIIII…

I’m guessing you view 123… as being half the size of 246…

If so, I don’t understand how you can do this considering the fact that B* is notan active progression without end. B* is a complete state. It IS Infinite in quantity. There cannot be more than one Infinity because there is no beyond Infinity in quantity. Infinity is a complete quantity. It is not a quantity that is being added to or can be added to. Nor is anything other than Infinity, the quantity Infinity. So it does not matter if I represent it as II… or IIIIIII… , the quantity is the same: Infinity. 123… is not half the Infinity that 246… is. They both denote the exact same quanity.

With A* you can make the argument that one is half the size of the other because both are finites that are actively increasing in size. Finites can increase in size potentially endlessly. Infinity cannot. Finites cannot become Infinite. The Infinite IS Infinite. Nothing can be added to it. Nothing can be taken away from it. It cannot be divided or multiplied. III… minus/plus I is absurd because it does not result in III… nor is it the same as III… So how will there be any room/possibility for another Infinity in terms of cardinality?

The reasoning is super simple and I am sure that you, wtf, are aware it.

It’s true that some mappings from the naturals to the evens are bijections and some aren’t. The definition of same cardinality says that if THERE EXISTS even one single solitary lonely bijection between two sets; then those sets are defined to have the same cardinality. It’s like the guy who robs a bank and is then called a bank robber. It does him no good to say, “Well, there were thousands of days on which I did not rob any banks.” Makes no difference. If he robbed one bank even once, he’s labeled a bank robber.

In our case, there is at least one bijection between the naturals and the events. Makes no difference that there are other functions between them that aren’t bijections.

Ok. Then we’re in agreement. There’s at least one bijection between them. Many other functions between them are not bijections. But like the bank robber, if there exists one bijection between them, we say they have the same cardinality.

And, I do agree, the evens are nevertheless a proper subset of the naturals. So we see that the naturals are in bijective correspondence with a proper subset of themselves. Making the naturals a Dedekind-infinite set.

That is the standard mathematical meaning, yes. Bijection = one-to-one correspondence. They have the identical meaning. Bijection is generally preferred, because a “one-to-one function” is an injective function – no two inputs go to the same output. But with this terminology, a one-to-one function is not the same as a one-to-one correspondence. So we say bijection in order to clearly disambiguate this slightly confusing terminology.

But bijection and one-to-one correspondence mean exactly the same thing.

I don’t see the problem. We agree on two things:

There is a bijection between the naturals and the evens, which means exactly the same thing as saying there is a one-to-one correspondence between them; and

The evens are a proper subset of the naturals.

This is something about infinite sets that we must get used to. An infinite set can generally be placed into bijection with one of its proper subsets. And since that is the case, Dedekind had the insight of making that property the very definition of an infinite set.

In Frege’s formulation, a set is the extension of a predicate. P(x) = “x is a horse” gives us the set of all horses; P(x) = “x is a set” gives us the set of all sets. This leads to a contradiction

In ZF, we adopt the rule that in order to specify a set via a predicate, we must first have an existing set. So if P(x) = “x is a set,” then we can form the set “X such that P(x)” where X is already known to be a set. This avoids Russel’s paradox.

The idea that to use a predicate to form a set we must start with some other set already known to exist, is captured in the axiom schema of specification. It’s an axiom schema, and not just an axiom, because it actually stands for infinitely many axioms, one for each predicate. In passing, we observe that ZF actually has infinitely many axioms because of this.

Your assumption already leads to a contradiction.

I didn’t look at this in detail, but you should review the proof of Russel’s paradox. My meta-argument is that Frege himself got the point instantly, and said so in print.

I don’t know what you mean by “encompass.” It’s true that in PA the collection of all numbers is not a set. But you CAN call it a collection, or an assemblage, or a gathering. You just can’t call the entire collection a set, because sets are very particular technical things and PA doesn’t have sets.

But if you want to contemplate the collection or class of all numbers, or of all sets, there’s no problem with that.

The set of all triangles (in the plane, say) is provably a set. A given triangle is uniquely characterized by a set of three non-colinear points in the plane. The set of points in the plain is a set; and the set of all triples of non-colinear points is a set. So the set of triangles is indeed a set.

On the other hand the notion of the set of all sets leads immediately to a contradiction.

Is it possible that the misunderstanding is on your part and not on the part of all the logicians and mathematicians of the 20th century?

Let me reiterate this point: There most definitely is a COLLECTION of all sets; or a CLASS of all sets. There just isn’t a SET of all sets, because sets are constrained by the axioms of set theory; and the set of all sets is inconsistent with those axioms. Russell proved it and Frege got the point immediately – right on the eve of publication of his great work on logic and set theory, as you know. Poor Frege, but he was intellectually honest enough to admit his mistake. You should carefully review your own thought process, because you are misunderstanding Russell’s paradox.

It’s not hypothetically impossible that the misunderstanding is on their part.

How? I am FULLY aware of Russell’s paradox. This is one area I would describe myself as being very competent in.

I’m aware of this.

It is true that in PA, the collection of all numbers is not equal to ALL numbers. Even if we call PA a collection, it is not a complete/finished collection. It is an expanding one. One that will never encompass/include ALL numbers (in the exact same way to how a set that is not a member of itself, cannot encompass/include/contain ALL sets that are not members of themselves).

That’s like saying you cannot have a set/collection of all existing things (which is clearly contradictory). Let’s use collection instead of set:

Let’s say Existence is the set collection of all existing things. There are other existing sets collections besides Existence (a set collection of cars for example). They ALL exist in Existence. Since Existence is the set collection of ALL existing things, it follows that the semantic of ‘Existence’ encompasses/includes all existing sets collections (including Itself). Have a look at my proof if you find think it contradictory to say “Existence is not the set collection of all existing things”.

Ok. I thought you were arguing against it. You seem to be.

I don’t know what that means. PA doesn’t talk about the collection of all numbers.

Perhaps you’re getting at the distinction between actual and potential infinity. Aristotle made a big deal about this. It’s more meaningful in philosophy than in math. The terminology doesn’t come up in math. And what of it, anyway? I don’t see the point you’re trying to make. If you want to say that PA gives us each of the numbers but no “completed” set of them, I’m perfectly aware of that and I agree. That’s why ZF includes the axiom of infinity, so that we can talk about the set (\mathbb N) of all natural numbers. This is all straightforward.

Myself, I have no trouble conceiving of the collection of all the numbers given by PA. I don’t think of them as coming into existence one after the other. They all exist at once. How do I know? Well if 0 exists then 1 exists; and if 1 exists then 2 exists; and so forth. That’s one of the rules of PA. So if you ask me if 4958348054935803 exists, I know it does … because 4958348054935802 exists (the number one unit less). And that exists because ITS predecessor exists; and I can get back to 0 in a finite number of steps. So all the numbers exist at once. There’s just no “set” of them, because PA doesn’t talk about sets. For that we need ZF. Just so we can talk about sets.

I have no idea what you mean by Existence. I do math, not metaphysics. I can’t help you with Existence-with-a-capital-E.

But if Existence is everything, then I’ll agree that it includes all the sets. But then it can’t itself be a set, while still conforming to the other axioms of set theory. Because that leads to a contradiction.

I will agree with you that the class or collection of everything includes itself, if you like. But that class or collection simply can’t be a set, because the rules of set theory are inconsistent with a set of all sets. “Set” is a technical term. Saying something is a set puts great constraints on it. It’s not at all synonymous with collections or classes.

A set is NOT what they tell you in high school, a “well-defined collection.” It’s nothing of the sort. In math, arbitrary collections are not sets; and the elements of sets need not be well-defined.

Bijections are about the size of the set, not the higher numbers involved.

A = {1,2,3,4}
B = {10,20,30}
A is larger than B

So while you are counting those two increasing sets, they remain the exact same size until you get to the end - and that is what is fooling most people because they know there is no end.

It seems laudable to say, “because the count in each set remains the same forever (an end is never reached) the size of the sets remains the same - forever”.

That sounds right doesn’t it? That is what wtf is accepting (with a great many others).

But it isn’t right.

I have shown you several cases of having more than a simple infinity (an infA).

N = {1,2,3…} = infA in degree/size
C = {N,a,b,c} = infA+3 in degree/size

N is said to be countable because it is the same as natural numbers that are used in the counting. But if we have ALL of N plus 3 more, we have used ALL of the counting numbers up yet we need to count up 3 more times.

C is NOT countable - it is too big to be counted (not enough numbers to cover all of the elements).

And that is why there is no bijection between them even though they are both infinite sets.

Russell asks, is the set of all sets that are not members of themselves, a member of itself? If no/yes, then… I know exactly what his point was, and I am against it for the following reasons:

You cannot have a set that is not a member of itself, be the set of ALL sets that are not members of themselves, precisely because it cannot be member of itself (it will always be one set that is not a member of itself that is outside the set of ‘all’ sets that are not members of themselves) So that definitely rules out the answer “no” with regards to the question he put to Frege.

The answer to his question is yes. x (the set of all sets) is the ONLY set that contains ALL sets that are not members of themselves (as well as itself, hence why the answer is yes). What contradiction/paradox to you see in this?

What Aristotle describes is PA and actual Infinity is ZF. The former is not infinity in any way. The latter is Infinity. I believe we are in agreement on this.

So you agree that it includes all the sets. What’s holding you back from describing it as being that set, that includes/encompasses all the sets?

If I say to you all triangles are meaningful shapes, I am saying to you that the meaning triangle is a member of the meaning ‘shape’ and the meaning ‘meaningful’. Consider the meaning ‘meaningful’. All meanings are members of the meaning ‘meaningful’ (because they are all meaningful). But the meaning ‘meaningful’ is not a member of any other meaning other than itself. Acceptance of Russell’s paradox implies that we accept the following: Not all meanings are members of the meaning ‘meaningful’. We do not accept this.

If you don’t mind, I am not going to engage with regard to Russell’s paradox. It’s a 120 or so year old result accepted by all mathematicians and logicians. Its logical structure is impeccable. It’s not productive for me to argue with known truths. I hope in time you come to accept the argument, but I can’t say any more about it than Russell himself did.

That’s been my understanding, but it’s not a good distinction. There is nothing potential about the collection of numbers 0, 1, 2, 3, 4, … There is no time involved. If 0 exists “in this moment,” then 1 exists in this moment, and 2 exists in this moment, and for any number n, n exists in this moment. Each and every one of the infinitely many numbers exists in this moment.

The potential/actual distinction may have been important to Aristotle back in the day, but it has no relevance or meaning in math, because there is no time in math. The sequence 0, 1, 2, 3, … doesn’t “become” one by one. Each number exists immediately as soon as you write down the rules:

0 is a number; and

If n is a number, so is n + 1.

In PA we can’t form a set of them, because PA doesn’t talk about sets. In ZF we can model PA via the axiom of infinity, and we can loosely say that “ZF contains a copy of all the natural numbers.” But the infinity of PA and the infinity of ZF are the same; except for the fact that there are no sets in PA so we can’t talk about the set of all numbers.

But it is, as I’ve just argued. There are infinitely many members of the sequence 0, 1, 2, 3, 4, …

No. There are infinitely many numbers in PA; and there is a model of PA in ZF, as well as a set containing all the elements of PA.

“It” here referring to Existence. I’ll stipulate that Existence contains all sets. The sets that contain themselves, the sets that don’t contain themselves. All the sets. I’m perfectly fine withthat.

Russell’s paradox, which shows that the set of all sets is inconsistent with the rules of set theory. “Set” is a loaded term, it comes along with some rules, and these rules are inconsistent with a set of all sets.

No you have this all wrong. “Set” is a loaded technical term. You wouldn’t say that the collection of all sets is a fish, because the collection of all sets doesn’t obey the rules we’ve agreed on that define a fish. Likewise the collection of all sets can’t be a set, because that collection is inconsistent with the rules we’ve agreed on for sets.

You’ve swapped out the word “set” and replaced it with “meaning.” But that’s invalid. “Set” is a loaded technical term. It implies conformance to the rules of set theory. “Meaning” does not have to conform to the rules of set theory.

You want “set” to mean any arbitrary collection, but that is not what a set is. That’s the high school or naive definition of a set. But a set is anything that obeys the rules of set theory; and those rules are inconsistent with a set of all sets.

Right, that’s your opinion. You think there’s a one-to-one correspondence between the set of naturals and the set of evens. I am perfectly aware of that – I know what you think – so this part of your response is unnecessary. What you were supposed to do is address (i.e. find the flaw in) the argument that I presented. Let’s see if you did that in the rest of your response.

But the conclusion of my argument is that there is no one-to-one correspondence between the two sets.

Alright, let me restate the entire argument.

Two sets are equal in size if and only if they can be put in one-to-one correspondence.

Two sets (A) and (B) can be put in one-to-one correspodence if and only if we can pair the elements of (A) and (B) such that every element in (A) is paired with a single element in (B) and every element in (B) is paired with a single element in (A).

The set of naturals contains every element of the set of evens plus some more.

From (3), we can deduce that whenever and however we pair every even number with a unique natural number, we are left with an infinite number of unpaired naturals.

From (2) and (4), we can deduce that there is no one-to-one correspondence between the set of naturals and the set of evens.

From (1) and (5), we can deduce that the set of naturals and the set of evens are not equal in size.

That’s my argument. It is your job to find the flaw in it. So far, it doesn’t look like you did that.

What’s wrong with it?

And remember that deductive arguments are faulty either because they are logically invalid (i.e. the conclusion does not follow from the premises) and/or because some of the premises are false.

What happens to be the case here?

You seem to agree with (1) and (2). You also seem to think that “bijective function” and “one-to-one correspondence” mean one and the same thing. If that’s the case, the above argument also shows that there is no such thing as a bijective function that maps the set of naturals to the set of evens.

What’s obvious to me is that there exists a mathematical expression such as (f(x) = 2x) that represents certain concept. The question is: what kind of concept does it represent? It’s obviously a function, and in mathematics, whether correctly or incorrectly, functions are understood as sets of ordered pairs ((x, y)) consisting of elements (x) belonging to the domain of the function and corresponding elements (y) belonging to the codomain of the function. If that’s what a function is, and if my argument is sound, then (f(x) = 2x) represents a conceptually impossible function. In other words, it’s an oxymoron, similar to that of a square-circle (which, of course, does not make it a useless concept.)

Well me, and all of the mathematicians and logicians in the world. Not that this proves anything, but at some point it’s unproductive for me to continue to argue with people about it. After a while it’s like arguing with flat earthers. It’s mildly entertaining, but only up to a point.

Well you’ve been making the same fallacious arguments for the several years I’ve known you, why is that ok? By your own logic, you needn’t bother articulating your own arguments, since I know what you think and I’ve heard your arguments many times already.

I have done so, many many times.

LOL. Ok!

Bijection and one-to-one correspondence are synonymous, as I and Wiki and every mathematician and logician in the world agree.

Ok.

This is a definition, and not a fact. We SAY two sets have the same size (with respect to cardinality) if there is a bijection between them. They may have the same size with respect to cardinality yet not with respect to the proper subset relation, or measure, or asymptotic density. You agree?

It’s like saying that two people have the same size when they have the same height, then noting that their weight differs, and claiming that “size” is contradictory. But all that shows is that when we use the word size, we have to say what we mean in a given context. We can compare the size of sets by cardinality, or measure, or proper subset relation, or asymptotic density, or probably lots of other ways I don’t even know about. The word size is not contradictory. It’s ambiguous, and may refer to many different things depending on context.

Yes, that’s the definition of one-to-one correspondence.

Undoubtedly true.

No, we can’t deduce that. “Whenever and however?” No. If we pair the naturals and evens via f(n) = 2n, we get a one-to-one correspondence. (4) is falsified.

If you deny that f(n) = 2n is a bijection or one-to-one correspondence, you must product a natural not mapped uniquely to an even, or vice versa. Can you do that?

(4) is falsified, therefore so is (5) which depends on in.

(5) is falsified because it depends on (4) which is falsified, so now (6) is falsified. But again, you’re equivocating the word size. Two people can be the same size with respect to height but not weight. Two sets can be the same size with respect to cardinality but not with respect to the proper subset relation, or with respect to measure, or with respect to asymptotic density. The word “size” is heavily overloaded with mutually inconsistent meanings. We need to be very clear which sense of the word we mean in any given context.

(4) is false hence (5) is false hence (6) is false. Your argument is valid but not sound, since your premise (4) is false. The evens are a proper subset of the naturals; nevertheless there is a bijection or one-to-one correspondence between them, namely f(n) = 2n

It’s valid but unsound. (4) is false.

Your premise (4) is false so your argument is valid but not sound.

Your argument is valid but unsound, being based on a false premise, aided and abetted by equivocation of the word “size.”

I think I’ve made my point. And if not, I at least have mainstream math and logic on my side. Not that this means much, but I am surely at the point of diminishing returns in this discussion.

A straight line through the origin of slope 2, as is taught in high school analytic geometry.

Yes, all true.

Correct. Therefore since f(x) = 2x is a perfectly valid function, your argument is unsound. You just proved it!

Surely you are not arguing that you can’t multiply a number by 2. Are you?

By the way the unit circle in taxicab geometry is a square. I wish people would stop using this example, since there are square circles. There’s a picture of one on the Wiki page I linked.

That’s up to you. If you don’t want to discuss any further, that’s fine by me. No justification needed, you can simply bail out. And you can do it at any time.

But that does not make the response that you gave me an adequate one.

Well, if you can pull them from your memory in their entirety, then there is no reason for me to repeat them.

But you didn’t do so in your previous post. And I don’t really think you ever did. If you think otherwise, you can give me a link or you can simply do it again.

In that case, my argument also shows that there is no bijection between the two sets.

Right. That’s a definitional premise.

My argument is unconcerned with other notions of size.

Alright, so you have no problem with (1), (2) and (3).

But you disagree with (4). You think it does not follow. If I made a mistake, where exactly does it lie?

The set of naturals contains every element of the set of evens plus some more. You agree with that. Thus, when you map every even number to a unique natural number, you use up all even numbers without using up all natural numbers. What exactly is wrong with that?

You are supposed to explain the mistake made in the argument, not “falsify” its conclusion by presenting a counter-argument. (This is what people kept doing in the first 20 pages or so of this thread. Instead of addressing the arguments presented in the OP, they kept presenting their own counter-arguments.)

Suppose I presented the following argument:

All men are mortal.

Godzilla is mortal.

Therefore, Godzilla is a man.

By quoting Wikipedia as a proof that Godzilla is not a man, you’d be presenting your own counter-argument instead of explaining what’s wrong with my argument.

That sort of behavior is unacceptable, if you ask me. But such rules aren’t standardized. Few, if any, abide by them.

How do you know that (f(n) = 2n) is not a contradictory concept that happens to be useful?

The fact that a mathematical expression such as (f(n) = 2n) exists, that certain concept is attached to it and that it happens to be useful does not mean that (f(n) = 2n) is not a contradiction in terms – an oxymoron.

How so?

Actually, (4) is a conclusion that is also used as a premise. So you can’t just say that it is false. You have to say what’s wrong about the way I arrived at it.

If f(n) = 2n is not a bijection between the naturals and the evens, you need to name a natural not mapped uniquely to an even; or an even not mapped uniquely to a natural. Can you do so?

0 1 2 3 4 5 ...
0 2 4 6 8 10 ...

You can see from the picture that every number on the top row is mapped uniquely to one on the bottom, and vice versa. If you claim otherwise, what number is not mapped?

Galileo made the exact same point in 1634 or so regarding the perfect squares. And this point was noted by Arab mathematicians in the 1200’s, and by the ancient Greeks thousands of years ago.

0 1 2 3 4 5 ...
0 1 4 9 16 25 ...

Was Galileo mistaken?

If I claim f(n) is a bijection and you claim it is not, you have to name a number not mapped.

Alright, so instead of addressing my argument, you decided to present yet another counter-argument (: That’s not exactly acceptable but I’ll address it anyways.

Logic dictates that you cannot put (A = {1, 2, 3, \dotso}) and (B = {2, 3, 4, \dotso}) in one-to-one correspondence. One element from (A) must be unpaired. That can be any element – it all depends on how you pair them. It can be (1) or it can be (2) or it can be (3) and so on. What you can do is you can come up with a concept that stands for something that is logically impossible. The concept of a square-circle is an example. And the same seems to apply to the concept of (f(x) = x + 1). This expression is DEFINED to stand for a one-to-one correspondence between (A) and (B). In itself, it is not a proof that such a correspondence is logically possible. And it seems that it is not. (f(x) = x + 1) represents an impossible bijection. So asking me a question such as “But which element isn’t paired?” misses the point. Of course, since (f(x) = x + 1) represents one-to-one correspondence between (A) and (B), what it stands for is NOT a relation that is not a bijection. In other words, there are no unpaired elements. But the relation it stands for is nonetheless impossible.

It makes no difference to the core issue at hand. You are looking at which set contains more numbers (in which case B is smaller than A), I was looking at which set adds up to contain the highest finite number (in which case B is bigger than A).

My point is that A and B are both finite sets that are growing in size endlessly. This means that they have an end or a finite size that they are constantly surpassing and increasing in. They will never amount to infinity in quantity, size, or even in sum total. Going back to the distinction between 123… and 123…

Quantitatively speaking, what’s the difference between:
IIIIII… and IIIIIIIIIIII…

Alright, I’ll try a different approach to convey what I’m trying to convey, but it does require an open mind and a sincere effort to grasp:

If I say to you ‘triangle’, then I am referring to all triangular things. The meaning of triangle is not an actual triangular object(exactly like how the set of triangle, is not an actual triangular object). If I say to you the set/meaning of ‘triangle’, then I am highlighting all triangular things. Meanings are the meanings that they are, and sets are the sets that they are (which at this point makes it look as though all sets are members of themselves, just as all meanings are members of themselves)

Tell me which step you disagree with:

Set means set and meaning means meaning. Set cannot mean something other than set. Agreed?

All meanings are meanings/meaningful. All sets are sets. Agreed?

The meaning/set of triangle is a member of the meaning/set of shape. Agreed?

There is no meaning/set that is not the meaning/set that it is (as in all meanings mean what they themselves mean). Thus, all true meanings/sets are at least members of themselves.

ONLY the meaning/set of ‘meaningful/all sets’ contains ALL meanings/sets as members of itself. If disagree, why?

The meaning of meaningful is ONE meaning, the set of all sets, is ONE set. Since there is no meaning/set that is not the meaning/set that it is, Russell’s question (and therefore paradox) becomes irrelevant, and we have a set/meaning that contains all sets/meanings as members of itself. It is the only set/meaning that is EXCLUSIVELY a member of itself. Whichever way you look at it, the answer to Russell’s question is a definitive no.