Hilbert's Hotel and Cantor's Diagonal Argument

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Hilbert's Hotel and Cantor's Diagonal Argument

Postby Positor on February 2nd, 2016, 10:36 pm 

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.)
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