On Sunday, 23 February 2014 19:21:35 UTC+2, Julio Di Egidio wrote:
> Indeed. To reiterate, mine was not an objection to the notion of 1-to-1 > mappings, e.g. I have no qualms with the idea that there are as many even > natural numbers as there are natural numbers: both collections are simply > *endless*. It is the idea that the natural numbers form a set, i.e. a > complete totality that is, as I am contending, untenable.
Can you really say anything about the cardinality of an infinite set? 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. It is Cantor's illogical ideas that have taken root like a cancer in mathematics. If you think that my analogy is bad, then consider that academic baboons will tell you every real number is mapped to a point on the number line. Of course there is no such thing as a real number but that is another topic.
> The consequence, again, is that Hilbert's hotel is illogical and, in fact, > misleading: either there is always room left, or all rooms are occupied, but > not both.
Yes. It's not even a paradox of any kind. In order to be a paradox, the conclusion reached must stem from a valid argument, not Jewish fables.
>IOW, if an infinity of rooms is occupied by an infinity of guests, there is no >room left, and the trick of moving each guest to the next room cannot be >performed: slightly more rigorously, if we have 1, 2, 3, etc. up to >(allegedly) *all* natural numbers, there is just no natural number missing >that we can add to the lot.
Do you honestly think there is such a thing as an "infinity of guests"? Can you reify any of this nonsense?