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.