I suggest we call a spade a spade. A falsity is a falsity. A conclusion derived from a false premise is unsound. An unsound argument does not constitute "a proof".

I suppose you could argue that mathematicians produce their own rules, and are not subject to the terms of logic. But what would be the point in giving mathematics such an exemption, to proceed in an illogical way. It seems like it would only defeat the purpose of the pursuit of knowledge, to allow for an illogical form of logic. — Metaphysician Undercover

I would agree with you if the object of this discussion were 'real' infinity as a 'real-world phenomenon'.
I find this 'real' infinity uncomprehensable, and so any speculation about it's properties, seems, well, at the very least, dubious. But this is not the case, as this thread concerns mathematical infinity. You're absolutely right - I argue that mathematicians set their own rules. Doesn't mean those rules can't eventualy change, of course, as paradigm shifts have occurred in other disciplines.

One of my professors used to say that pragmatism is not a philosophy at all. So perhaps a pragmatic stance on this question is not philosophical. Still, I would argue that if the 'orthodox' view of mathematical infinity solves more problems than it creates, then so be it.

1) To say that S is larger than S' means that S' is a proper subset of S.
( A definition that applies to all sets, regardless of their size. ) — Magnus Anderson

This is false, since that definition applies only to finite sets. — Banno

It doesn't even work for finite sets. Think what it would mean if you could only compare the sizes of sets and their subsets. You couldn't say, for example, that there are more apples than oranges on the table, because neither is a subset of the other.

Magnus is right in spirit, but isn't referring to natural numbers, but to "lawless choice sequences" that are infinite yet Dedekind-finite, meaning that the sequences are of finite but growing length.

By contrast, the naturals are "lawful" choice sequences, which by construction are essentially dedekind-infinite functions that don't represent sequences in the flesh, and are what a type-theorist would say are purely intensional sequences that shouldn't be confused with actual sequences.

To rectify an earlier confusion, the computer-science meaning of "extension" refers to explicit data. According to this definition, the identity function on the naturals ( \lamda (n : N) => n ) is an extension in the sense of a function, whereas the graph of that function, namely the set { (n,n) | n is a Natural number} isn't an extension. But confusingly for philsophy that graph is considered an extension according to the Fregean notion of extension, since Frege defined an extension as referring to the arguments of a predicate that make it true.

In effect, Frege conflated the notion of data-at-hand with the notion of functions that can produce data on demand, as a result of thinking that functions exist independently of their domains and ranges. For Frege, and unlike the computer scientist, a function isn't a causal operation that transforms input into output, but a transcendental relation that relates a static domain to a static range. Hence Frege interpreted predicates (which he called "concepts") as being non-destructive testers of their domains, which naturally implies that concepts and Fregean extensions exist independently and in one-to-one correspondence, leading to Russell's Paradox and also led to the failure of formalists like Hilbert to predict incompleteness.

There's no need to list all of the elements. All this talk about constructivism, intuitionism and finitism misses the point ( I do not subscribe to any of these -isms nor do I have to in order to be internally consistent. )

PROOF

1) To say that S is larger than S' means that S' is a proper subset of S.
( A definition that applies to all sets, regardless of their size. )

2) N is a proper subset of N0.

3) Therefore, N0 is bigger than N.

This is an indisputable proof. As indisputable as 2 + 2 = 4.

However, if you're convinced by a fallacious proof, you will normally deny the validity of this one, like a cancer attacking healthy cells.

FALLACIOUS PROOF #1

The first fallacious proof they use to show that N and N0 are of the same size is the observation that, if you add 1 to infinity, you still get infinity. This is true but only in the sense that the result is also an infinite number ( i.e. larger than every integer. ) They make a mistake when they conclude that, just because "infinity" and "infinity + 1" are infinite numbers, it follows that they are equal. It's like saying that 4 equals 5 merely because 4 and 5 are integers. — Magnus Anderson

What else would this be than finitism?

You don't accept the infinite to be different from the finite and obviously treat infinite like it would finite by arguing that "infinity" and "infinity + 1" aren't equal. Just look what the axiom of infinity is, which @Magnus Anderson clearly thinks is incorrect. That n < n+1 is simply how finite numbers work.

It doesn't even work for finite sets. Think what it would mean if you could only compare the sizes of sets and their subsets. You couldn't say, for example, that there are more apples than oranges on the table, because neither is a subset of the other. — SophistiCat

I think that @Magnus Anderson seems to think that if you take one out of an infinity set then number 1 is really missing from there.

Sorry, Magnus, but your "proof" merely begs the question. All you have done at this point is:

• asserted impossibility without derivation
• treated definitional existence as illegitimate by fiat
• accused others of fallacy and bad faith for not sharing your standards
• refused to specify what would count as proof

This is why the discussion keeps looping. If you want to move the discussion forward you need to either (1) derive (not assert) an actual contradiction within the accepted mathematical framework (per ) or (2) reject the standard framework and present a coherent alternative (e.g. intuitionism, finitism, non-classical logic, etc.).

As it stands, Banno has already shown that combining your premise (1) with transitivity, antisymmetry and the existence of infinite partitions leads to contradictions. At this point there is nothing of substance left to discuss.

At this point there is nothing of substance left to discuss. — Esse Quam Videri

Yep.

And Banno, you were right.

Rather what the OP specifically referenced, which is the infinite numbers between infinitely minute numbers. — LuckyR

I think this matter still has relevance. It is the issue of division. In reality, everything that we attempt to divide can only be divided according to its nature. Nature dictates the way something can be divided. We cut things up very evenly using instruments of measure, but eventually we get to molecules and then atoms, and we are greatly restricted in our capacity to divide "evenly". However, some things like space and time, we might not find the natural restrictions, and so we would be inclined to apply principles of infinite divisibility. Since the mathematical principle of divisibility (infinite) does not correspond with the real divisibility of the substance (space and time), the uncertainty principle is produced.

I would agree with you if the object of this discussion were 'real' infinity as a 'real-world phenomenon'.
I find this 'real' infinity uncomprehensable, and so any speculation about it's properties, seems, well, at the very least, dubious. — Zebeden

Infinity is a "real-world" phenomenon. We have examples of it as the infinite decimal extension of pi, and of the square root of two. The circle, and the square are extremely useful real world applications, yet the principles which validate their use lead us into these real-world infinities.

We might dismiss the problem by saying there is no such thing as a true square, or a true circle, in the real world, and dismiss these conceptions as ideals without real world validation, but that doesn't resolve anything. It just produces a division between conceptions and the real world, where we allow ourselves to employ false premises for the sake of usefulness, and we lose the epistemic value of "truth". Truth is no longer a requirement for knowledge, and we allow that we are not guided toward the truth.

Instead, we ought to look at these issues, where the ideal does not correspond with the real world, as demonstrations which show where our ideals have been compromised by selecting usefulness over truth. They display where our understanding of reality faulters, as reality is fundamentally different from how we represent it. If you just say "I don't care about the true nature of reality, if the principles serve the purpose that's good enough for me" this is a violation of the philosophical mindset which seeks truth. And if we're always happy with the way things are working now, knowledge never advances.

Still, I would argue that if the 'orthodox' view of mathematical infinity solves more problems than it creates, then so be it. — Zebeden

This is not a good standard because the comparison cannot be made. The problems which are solved can be pointed to and numbered. The problems created are associated with the unknown and cannot be counted, nor can the extent or size of the problems be determined. The resolved problems are finite, the created may be infinite and uncountable. So, for example, we created CFCs, and that resolved a whole lot of different problems which we could point to. However, at that time we didn't know what was going on with the ozone, and we couldn't compare the created problems. This is the issue then, the problems created are hidden within the unknown, and only when they start to fester do we take them seriously, and seek out their depth and roots. The example I use above, which displays the problem of unruly use of infinity is the uncertainty principle. We don't know what is hiding beneath that name.

This is why the discussion keeps looping. If you want to move the discussion forward you need to either (1) derive (not assert) an actual contradiction within the accepted mathematical framework (per ↪Banno) or (2) reject the standard framework and present a coherent alternative (e.g. intuitionism, finitism, non-classical logic, etc.). — Esse Quam Videri

1. The actual contradiction is blatant, and I've stated it.
2. Rejection of the framework because it is contradictory and false, is the task of philosophers. Presenting a coherent alternative is the task of mathematicians. Therefore you are wrong to suggest that the one who refutes the framework is obliged to present another.

At this point there is nothing of substance left to discuss. — Esse Quam Videri

The problem is clear. The mathematicians in this forum refuse to accept the refutation, though it is very sound. Because of this, they refuse to get on with the task of producing a coherent alternative. For the philosophers, "there is nothing of substance left to discuss", because the refutation is clear, and the mathematicians remain in denial. Until the mathematicians accept the refutation, and start again at the foundation, the philosophers will have nothing to offer, and there will be nothing of substance to discuss.

Sorry, Magnus, but your "proof" merely begs the question. — Esse Quam Videri

You would have to prove that. I am, however, pretty sure you can't do it. But I can show, as I already did, that YOUR reasoning is circular. So what you're doing here would be sort of like a projection.

asserted impossibility without derivation — Esse Quam Videri

Again, you're doing the very thing you accuse me of. You're asserting something without proving it.

I presented a very clear process of derivation.

treated definitional existence as illegitimate by fiat — Esse Quam Videri

The onus of proof is on the one making the claim. You have to show that just because you can define a symbol as a bijection between N and N0 that it follows that such a bijection exists. I can show you why that does not follow. And I kind of already did.

accused others of fallacy and bad faith for not sharing your standards — Esse Quam Videri

Not for sharing my standards but for not being independent thinkers ( which is fine ) while pretending that they are ( which is not fine. ) Not everyone is an independent thinker -- and does not have to be.

refused to specify what would count as proof — Esse Quam Videri

Actually showing that bijection between N and N0 exists by employing definitional logic. So far, you've been merely asserting it and relying on something that is very much like a circular argument.

This is why the discussion keeps looping. — Esse Quam Videri

That's not the real reason. The real reason is that people do not know how to think outside of the box. People are missing the point all over the place. The main point of dispute is never addressed.

derive (not assert) an actual contradiction within the accepted mathematical framework — Esse Quam Videri

I already did that. But if your argument is a non-sequitur, it's not really necessary to to do so, isn't it? All I have to do is to show that it's a non-sequitur.

As it stands, Banno has already shown that combining your premise (1) with transitivity, antisymmetry and the existence of infinite partitions leads to contradictions. — Esse Quam Videri

He hasn't.

At this point there is nothing of substance left to discuss. — Esse Quam Videri

There is. Quite a bit. But all that is left is to think. No more room for quotations. But non-thinkers don't think.

The problem is clear. The mathematicians in this forum refuse to accept the refutation, though it is very sound. — Metaphysician Undercover

I’m afraid it’s not, and I’ll try to clarify why.

All you’ve claimed so far is that mathematicians are working with a notion of infinity that you don’t accept, and you’ve given some philosophical reasons for rejecting it. That’s a legitimate philosophical position.

The problem is that this is a philosophical objection, not a mathematical one, and as such it doesn’t justify the claim that the mathematical notion of infinity is contradictory. The mathematical definition is perfectly sound relative to the formal system in which it is embedded.

By analogy: suppose we’re playing a game of Chess and, on your turn, you legally move your queen from d1 to a4. Suppose I respond to your move by saying: “that move doesn’t make sense because in real life kings are more powerful than queens and so only kings should be able to move like that”. That may be a fine external critique of the rules of Chess, but I haven’t thereby shown your move to be illegal. Given the established rules, it was a perfectly valid move.

Likewise, your objection to the mathematical notion of infinity is a meta-level objection. It doesn’t undermine the internal coherence of mathematics as it is standardly practiced. At most, it shows that the standard mathematical notion of infinity conflicts with your own metaphysical views.

If you wanted mathematicians to take this challenge seriously as mathematics, it would require proposing an alternative formal framework built around your accepted notion of infinity and showing that it does at least as much mathematical work as the existing one. As things stand, no such reason has been given for abandoning the standard definition.

I'll leave it at that.

Let me illustrate my point.

Functions can be malformed. They can contain internal contradictions that effectively render them as non-existent.

Consider the following example.

Let A be { 1, 2, 3, 4 }.

Let B be { 0, 1, 2 }.

Consider the function f: A -> B, f( n ) = n - 1.

Since this is a function, and since functions are relations where every element from the domain is paired with exactly one element from the codomain, this is also a contradiction. Being a function means 1) it must obey the rules, and 2) every element form A must be paired with exactly one element from B. But both are violated. The rules say that the number 4 should be paired with the number 3, but no number 3 exists in B. It being a function means that the number 4 must be paired with exactly one element; otherwise, it is not a function. But it isn't paired with any element.

So even though this function is defined as a bijection, no such bijection exists.

What this means is that you have to show that f: N -> N0, f( n ) = n - 1 is not a contradiction in terms before you can conclude that it exists.

Has anyone done that?

Of course not.

Instead, the opposite has been shown.

Let A be { 1, 2, 3, 4 }.

Let B be { 0, 1, 2 }.

Consider the function f: A -> B, f( n ) = n - 1.

Since this is a function, and since functions are relations where every element from the domain is paired with exactly one element from the codomain, this is also a contradiction. Being a function means 1) it must obey the rules, and 2) every element form A must be paired with exactly one element from B. But both are violated. The rules say that the number 4 should be paired with the number 3, but no number 3 exists in B. It being a function means that the number 4 must be paired with exactly one element; otherwise, it is not a function. But it isn't paired with any element. — Magnus Anderson

And is there an element n of N such that n-1 is not a member of N0?

This is a perfectly good argument, but it is not the argument you make about N and N0, which relies on the claim that if B is a proper subset of A, its cardinality must be smaller. Here no mention is made of cardinality.

To argue that f(n)=n-1 does not map every member of N to a member of N0, you must show, as you did here, that there is an n for which f(n) is undefined or not a member of N0.

To argue that f(n)=n-1 does not map every member of N to a member of N0, you must show, as you did here, that there is an n for which f(n) is undefined or not a member of N0. — Srap Tasmaner

Not really. And that's a commonly made mistake.

Bijection does not mean that you can take ANY element from N and uniquely pair it with an element from N0. Of course you can do that with N and N0.

It means that you can take EVERY element from N and uniquely pair it with an element from N0. And that's what you can't do.

Do you see the subtle difference?

And remember, the onus of proof is always on the one making the claim.

You can't claim that bijection exists between N and N0 merely because you can take any element from N and uniquely pair it with an element from N0. That does not follow.

Do you see the subtle difference? — Magnus Anderson

What is the cash value of that difference, as you see it?

The difference is that bijection means that you can take every element from N -- not merely any arbitrary subset of it -- and uniquely pair it with an element from N0.

You have to do it for all of the elements from N.

You don't know if that's possible. You just know that every element from N can be uniquely paired with an element from N0.

I see. I would say there's a difference between making a claim about "a subset" and a claim about "any subset"; many of us will treat the former as a "some" claim and the latter as an "all" claim. We similarly take "arbitrary" to imply "all" claims. Perhaps if we simply agreed on how we're using these words, there would be no dispute ...

Substantively, would you accept mathematical induction as showing that the mooted function maps every element of N to an element of N0? The proof is not hard.

Btw:

bijection means that you can take every element from N -- not merely any arbitrary subset of it -- and uniquely pair it with an element from N0. — Magnus Anderson

You have to do it for all of the elements from N.

You don't know if that's possible. You just know that every element from N can be uniquely paired with an element from N0. — Magnus Anderson

Is there also a difference between "all" and "every"? Because you seem to be granting what you denied ...

And Banno, you were right. — ssu

Things would be so much easier if everyone just accepted this dictum. :wink:

Thanks - your acknowledgement is appreciated.

Excellent use of the chess analogy.

While it's good to see you using some formal notation, this isn't an example of a function. A function from A to B is a set of ordered pairs satisfying certain conditions. So writing f : A → B, f(n) = n − 1 is not merely symbolic stipulation; it is a claim that the rule maps every element of A into B. But in this case, as you point out, not every element of A maps to an element of B.

That's not a contradiction within a function, as you diagnose, but a failure to specify a function.

That is, <f : A → B, f(n) = n − 1, with A ={ 1, 2, 3, 4 } and B = { 0, 1, 2 }> does not construct a function. It's ill-formed.

In the argument we are considering, there is no such malformation. This is not just asserted, but demonstrated by the conclusion that the function is well-defined, injective and surjective.

Unlike the finite example, f: ℕ → ℕ₀, f(n) = n − 1 does satisfy the totality requirement: every natural number n has n−1 ∈ ℕ₀. Therefore the function exists and is indeed a bijection.

What this means is that you have to show that f: N -> N0, f( n ) = n - 1 is not a contradiction in terms before you can conclude that it exists.

Has anyone done that? — Magnus Anderson

Well, yes.

In standard mathematics, we can define a function f: ℕ → ℕ₀, f(n) = n − 1, and check the definition. We saw that every n ∈ ℕ maps to exactly one element in ℕ₀.

Once the definition is satisfied, the function exists by construction. There is no need to “show it is not a contradiction.” A contradiction would arise only if the rule could not possibly assign outputs in the codomain, which is not the case here.

You suppose that before a function exists, we have to show it is not contradictory. But in mathematical thinking we define the function, check the definition, then if all requirements are satisfied, the function exists.

Again, it is up to you to show any contradiction, not up to us to show there isn't one. You have misplaced the burden of proof.

Cheers. :up:

Yep.

Bijection does not mean that you can take ANY element from N and uniquely pair it with an element from N0. Of course you can do that with N and N0.

It means that you can take EVERY element from N and uniquely pair it with an element from N0. And that's what you can't do.

Do you see the subtle difference? — Magnus Anderson

The proof given shows that for each element in $\mathbb{N}$ there is exactly one element in $\mathbb{N}_0$.

Take any element of $\mathbb{N}$ and there is a corresponding element in $\mathbb{N}_0$. Take any element in $\mathbb{N}_0$ and there is a corresponding element in $\mathbb{N}$.

Since that works for any element, it works for every element. There is no gap. The rule at work here is Universal Generalisation.

Therefore the sets are the same cardinality.

All you’ve claimed so far is that mathematicians are working with a notion of infinity that you don’t accept, and you’ve given some philosophical reasons for rejecting it. — Esse Quam Videri

That's not true. The definition of infinity I use is the one used in mathematics, to describe the natural numbers as unbounded, unlimited, without end. I do not reject this definition of "infinity".

The problem is that this is a philosophical objection, not a mathematical one, and as such it doesn’t justify the claim that the mathematical notion of infinity is contradictory. The mathematical definition is perfectly sound relative to the formal system in which it is embedded. — Esse Quam Videri

Again this is not true. The philosophical objection is based in a fundamental logical principle, the law of noncontradiction. I demonstrated that mathematicians employ contradiction when they claim that the natural numbers are countably infinite, or a countable infinity. By the mathematicians' own definition of infinite, or infinity, it is contradictory to say that an infinity can be counted because "infinite" means that we cannot have such a count, it could never be acquired.

By analogy: suppose we’re playing a game of Chess and, on your turn, you legally move your queen from d1 to a4. Suppose I respond to your move by saying: “that move doesn’t make sense because in real life kings are more powerful than queens and so only kings should be able to move like that”. That may be a fine external critique of the rules of Chess, but I haven’t thereby shown your move to be illegal. Given the established rules, it was a perfectly valid move. — Esse Quam Videri

This is not analogous. I clearly show how the move of the mathematicians is 'illegal' (to use your word) within standard rules of logic, because it is contradictory. The natural numbers are defined as infinite, meaning limitless, endless, impossible to count them all. Then they say the very opposite, that the natural numbers are countable. Clearly, "countable infinity" is a contradictory concept where the first term contradicts the second. These are not my definitions which I have made up for this purpose. This contradiction is within the way that mathematicians themselves define the terms.

Likewise, your objection to the mathematical notion of infinity is a meta-level objection. It doesn’t undermine the internal coherence of mathematics as it is standardly practiced. At most, it shows that the standard mathematical notion of infinity conflicts with your own metaphysical views. — Esse Quam Videri

Again, this is wrong. The incoherence is internal to mathematics. The notion of "infinity" used by mathematicians themselves, is contradicted by the predication they make, when they propose a "countable" infinity. Here's an example much better than your chess proposal because the chess proposal fails to capture the situation.

Lets say we have a concept called "unintelligible" (analogous in this example to infinite). Then, we notice that there are different sorts of unintelligible things, that things are unintelligible in a number of different ways Different sorts of infinities). So, instead of studying the reason for, and the difference between, the different ways that unintelligibility appears to us, we simply name one of the forms of unintelligibility the "intelligible unintelligibility" ( analogous to countably infinite). Then we proceed to compare the other forms unintelligibility to this, under the illusion (falsity by contradiction) that we have made this type of unintelligibility intelligible by naming it so.

That is what the concept of "countably infinite" does. It creates the illusion (falsity by contradiction), that this type of infinity is actually countable. It's far better to use a concept like "transfinite", and state that the transfinite are a special type of infinite, but maintain they are not countable. This would exclude the possibility of an infinite set, or a transfinite set as this is the mistaken venture. It is the attempt to contain the boundless, limitless (infinite) into a set which is defined as an object, that requires the employment of contradiction. Putting limits to the limitless is contradictory.

f you wanted mathematicians to take this challenge seriously as mathematics, it would require proposing an alternative formal framework built around your accepted notion of infinity and showing that it does at least as much mathematical work as the existing one. As things stand, no such reason has been given for abandoning the standard definition. — Esse Quam Videri

The standard definition of "infinite" is not a problem whatsoever. so there is no need to abandon it. The proposal of "countably infinite" is a problem.

I clearly explained why it is not necessary, and actually inappropriate for me to propose an alternative framework. If mathematicians do not understand that they have incorporated contradiction within their framework, and so they are not inclined to rectify this, then I will just keep pressing this point. Maybe they never will.

Excellent use of the chess analogy. — Banno

The analogy is not similar. I have shown that the internal rules of the game (mathematics) are contradictory. Unless noncontradiction is not a rule in the game (mathematics), then the analogy fails. Are you and Esse Quam prepared to take that stance, to insist that the rule of noncontradiction is not a rule in the mathematician's game? if so, you might be able to make the analogy work.

infinity is a useful lie.

While it's good to see you using some formal notation, this isn't an example of a function. A function from A to B is a set of ordered pairs satisfying certain conditions. So writing f : A → B, f(n) = n − 1 is not merely symbolic stipulation; it is a claim that the rule maps every element of A into B. But in this case, as you point out, not every element of A maps to an element of B.

That's not a contradiction within a function, as you diagnose, but a failure to specify a function.

That is, <f : A → B, f(n) = n − 1, with A ={ 1, 2, 3, 4 } and B = { 0, 1, 2 }> does not construct a function. It's ill-formed. — Banno

You very clearly don't understand how language, definitions and oxymorons work.

Square-circles aren't squares either. They are also not circles. But they are also squares. And they are also circles. That's why they are oxymorons. They are two opposite things at the same time.

The same applies to the function that I mentioned. It is a function. But at the same time, it is not. The notation implies both. Not merely the latter. That's the mistake you're making. When you write, f: { 1, 2, 3, 4 } -> { 0, 1, 2 }, f( n ) = n - 1, that implies a function.

Symbols are capable of containing contradictions. It's not a new thing. It's the basis for the law of non-contradiction. It's not the case that symbols are necessarily X or not X. They can be both.

But either way, that's not really important, and it's nitpicking at best. The point is that the function does not exist. Which you agree with. And what that shows is that, just because you can define a function, it does not mean it exists. That was the entire point of that post.

In the argument we are considering, there is no such malformation. This is not just asserted, but demonstrated by the conclusion that the function is well-defined, injective and surjective. — Banno

And that's not true.

The only thing that you have shown is that you can take any element from N and uniquely pair it with an element from N0.

In other words, you have shown that, if we randomly pick an element from N, we can find its unique associate. Let's say we pick 1,345,219. Its unique associate would be 1,345,218. That holds true for every element from N. There are no exceptions.

But that does not mean we can put the two sets in one-to-one correspondence. That's a different thing. You haven't shown how many elements from N can be uniquely paired with N0. And your job is to show that you can take as many elements as there are in N0. Have you done that? Of course not.

All you have shown is that you can take an arbitrary subset of N that isn't larger than N0 and put it into a one-to-one correspondence with a subset of N0.

Do you understand the difference between the two?

Again, it is up to you to show any contradiction, not up to us to show there isn't one. — Banno

The onus of proof is always on the one making the claim. If you're making the claim that bijection between N and N0 exists, you have to show it, and that means, you have to show that such a bijection is not a contradiction in terms. That's what it means to show that something exists in mathematics.

You haven't done that. But I have done otherwise ( contrary to what you say. ) But you won't that accept because you're overly attached to your fallacious proof -- essentially, a circular argument -- that there exists a bijection between N and N0. "We can take any element from N and uniquely pair it with an element from N0, therefore, there's one-to-one correspondence between N and N0."

Again, if all you're going to do is spend all of your time justifying your chosen authorities, which is precisely what you're doing, then you want see the mistake they are making.

Is there also a difference between "all" and "every"? Because you seem to be granting what you denied ... — Srap Tasmaner

It's a subtle point that is difficult to explain. Perhaps I should have used "any" instead of "every" in that second quote of mine.

Let me try to explain it another way.

You have a device that basically mimics the operation of f: N -> N0, f( n ) = n - 1. You can type in a natural number, starting at 1, to get a number from N0.

For one and the same input that you type in, you will get one and the same output from N0. Moreover, for every output from N0, there is exactly one number from N that can generate it.

That's what makes it a bijection in the weaker sense. And it's trivially true. We all agree about that.

But what that does not mean is that you can type in all of the numbers from N and produce all of the outputs from N0. That's a bijection in the stronger sense ( the one that matters. )

Suppose that you have all the time in the world. Suppose that means that the number of days at your disposal is larger than the number of natural numbers. Suppose that you decide to type in every number from N. On day 1, you type in 1. On day 2, you type in 2. And so on. How do you know that you will be able to produce all of the outputs? The fact that the function is bijective in the weaker sense does not tell you that.

Again, this is wrong. The incoherence is internal to mathematics. The notion of "infinity" used by mathematicians themselves, is contradicted by the predication they make, when they propose a "countable" infinity — Metaphysician Undercover

countable

It all depends on how one defines "countable"

and an infinite number of those infinities, and... (infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and...) continues forever. and that continues forever. and that continues forever. and that continues forever. and that continues forever. and.....(…)… — an-salad

Infinity is a concept saying that there is no end in counting (in math), or final point (in physics or the material word or movement). If there were the end point of counting or movement is reached, then it wouldn't be infinity. Hence it is just an abstract concept, which doesn't exist in the real world.

Trying to count or prove infnity using math formulas or functions on the concept sounds silly and obtuse.