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Topic: The self-contradictory infinite hotel
Replies: 62   Last Post: Feb 24, 2014 10:17 PM

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 LudovicoVan Posts: 4,165 From: London Registered: 2/8/08
Re: The self-contradictory infinite hotel
Posted: Feb 23, 2014 2:56 PM

"John Gabriel" <thenewcalculus@gmail.com> wrote in message
<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.

Julio