"John Gabriel" <email@example.com> wrote in message news:firstname.lastname@example.org... <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 if-then. 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.