Re: infinitely many nn's = infinite nn's?

G. Frege wrote:

On Mon, 05 Mar 2007 18:01:20 GMT, Phil <toob-headman@xxxxxxxxxxxxx> wrote:


I don't suppose you could come up with a logical train of thought that shows why we CAN mix-and-match potential and actual infinity?


Well, actually Cantor wrote something about that question!

His argument in a nutshell: The notion of potential infinity only makes
sense on the "basis" of [the notion of] actual infinity. (But even if he's
wrong concerning this question, clearly the acceptance of "actual infinity"
allows for arguments referring to "potential infinity". The problem is just
the other way round. :-)


Remember, so I don't waste YOUR time, I have no disagreement with either the argument proving infinitely many numbers, nor the argument proving they are all finite. It's the combination that I think violates the rules of mathematics,


It doesn't. There are infinitely many natural numbers, but each and any is
finite.

I'll agree that this MIGHT _sound_ strange. Maybe the following formulation
is easier to digest:

Each and any natural number is finite.

But the set of _all_ natural numbers is not bound.
(I.e. it does not have an upper boundary.) With other words, there is no biggest natural number. (I guess that's immediately clear, no? If W would be this number, W+1 would be another natural number, but bigger. Which
is an aburdity.)


F.

George, this looks useful and workable; I'll get back to it later!

Phil
.

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