Positor » February 2nd, 2016, 8:36 pm wrote:Hilbert's Hotel can accommodate any number of extra guests, even when full, because the additional guests do not increase the cardinality of the infinity of existing guests. However, Cantor argues that an array of infinite rows of digits cannot accommodate a diagonally altered string of digits, because the inclusion of the altered string increases the cardinality.

Now, suppose that Cantor's horizontal rows represent the ID numbers of the existing guests in Hilbert's Hotel, and the altered diagonal represents the ID number of a would-be additional guest. Can the would-be guest be admitted to the hotel? If not, can he/she get in by the simple expedient of changing his/her ID number to one of Cantor's

horizontal numbers (or to an

unaltered diagonal)?

(I presume that if we can postulate an infinite hotel and infinite rows of Cantorian digits, there is no problem about having infinitely long ID numbers.)

Thanks Alan for bumping this up. It's easily answered.

Of course Cantor's diagonal argument doesn't say that the anti-diagonal can't be added to the list. It can, via the standard Hilbert trick. All Cantor says is that the anti-diagonal can't possibly

already be on the list, and that's the case here.

So you take the anti-diagonal, which is not on the list, and you move each number in a given row of the array up one row. The number (or guest) in row 1 goes to row 2; the number in row 2 goes to row 3, etc. This leaves row 1 empty, and you put the anti-diagonal there.

Now Cantor points out that there's a

new anti-diagonal, which isn't on the list, but that can be put on the list.

The point of Cantor's clever argument is that the original list of reals was arbitrary. We can conclude that given

any list whatsoever, there's some real -- namely the anti-diagonal -- that's not on the list. So the original list was not complete. And since it was an arbitrary list of reals, we conclude that

no list of reals can contain all the reals.

Note by the way that this is NOT a reductio or proof by contradiction. On the contrary. We start with an arbitrary list and show it's not complete.