LudovicoVan
Posts:
4,025
From:
London
Registered:
2/8/08


Re: The selfcontradictory infinite hotel
Posted:
Feb 23, 2014 2:56 PM


"John Gabriel" <thenewcalculus@gmail.com> wrote in message news:ca6204446dc34e5db58d554505a436a8@googlegroups.com... <snip> > To say that there are as many even natural numbers as there are natural > numbers is not proven by a bijection. Here's an example: > > Suppose that the natural numbers correspond to points on a number line. > > 1 2 3 4 5 6 7 8 9 10 ... > > Now remove all the even points: > > 1 3 5 7 9 ... > > So you still think that there are as many even natural numbers as there > are natural numbers? A bijection proves nothing.
That is not about proof, cardinality in terms of bijections is rather a definition. That said, yes, I think it is sensible to say that the collection of even natural numbers is equinumerous to that of all natural numbers, but I have already characterised in which sense I mean that: in the sense that both are *endless*, nothing less and nothing more.
<snip> > Do you honestly think there is such a thing as an "infinity of guests"? > Can you reify any of this nonsense?
Again, just as much as there is an infinity of natural numbers: i.e. it is another ifthen. And yes, I don't think saying there is an infinity of natural numbers is senseless, I just think saying that these form a *complete* totality (a set, more specifically) is senseless.
Julio

