Re: abundance of irrationals!)

*** T. Winter said:
> In article <MPG.1ce2b9fc4535b69c989ba7@xxxxxxxxxxxxxxxxxxxxxxxxx> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
> > *** T. Winter said:
> > > In article <1115214755.743369.314730@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> ...
> > > > 2) Finitey of numbers does not imply finite sets. Then my proof holds
> > > > for infinite sets too. It shows that there are no infinite sets.
> > >
> > > Why? Your bijection does not show infinite sets, that is not a proof
> > > that infinite sets do not exist. If I give the following bijection:
> > > 1 <-> apple
> > > 2 <-> pear
> > > this does *not* show that there are only two kinds of fruit.
> > >
> > > > You can argue as you like. There is no outcome other than "I believe in
> > > > set theory. Amen"
> > >
> > > You can argue as you like. There is no outcome other than "I believe there
> > > is no infinite set of naturals. Amen."
> > >
> > ***, I really think you're being purposely dense here. WM's progression is
> > obviously the set of the first n natural numbers, which I think we can all
> > agree becomes the set of natural numbers as n reaches infinity.
>
> Nope. It will never become the set of natural numbers, because n will not
> reach infinity. If n reaches infinity there must be a step before that
> were it is not yet infinite, and by adding 1 we get infinity.
>
Well, by everyone's need to make the naturals a set of finite numbers, and my
claim that that means it's a finite set, then you should be satisfied with a
finite set of finite natural numbers. However, just because counting STINKS for
dealing with infinity, that doesn't mean we can't use other methods to talk
about what happens at actual infinity. For instance, if I take a finite number,
and divide it by two at each iteration, I feel pretty confident you can tell me
what value I have after an infinite number of such iterations, even without
knowing the finite number I started with, can't you?
--
Smiles,

Tony
.

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