On Saturday, December 7, 2013 2:26:54 AM UTC+1, Virgil wrote: > > On Friday, December 6, 2013 4:33:19 AM UTC-7, Albrecht wrote: > > > > > >> The diagonal argument from Georg Cantor is so silly and easy to > > >> destroy. Let's have a "antidiagonal" of all natural numbers. No > > >> problem: For any list of naturals we can take the first number and > > >> change it by adding 1 to gain d_1, take the second number and, if > > >> different from d_1, change it by adding 1 to gain d_2, else > > >> d_2:=d_1, and so on. > > >> > > >> For all numbers of the list, the (equivalent to Cantor's > > >> antidiagonal) gained antidiagonal d_list is natural (proof: a > > >> natural +1 is natural) and different from any number of the list > > >> (proof: it is at least 1 more than every number of the list by > > >> construction) and thus the list isn't complete. Since this holds for > > >> _any_ list of naturals --> the naturals are uncountable. > > > > Albrech totally misunderstands the canto argument. > >
Virgil totally misunderstand my arguing against the second diagonal argument of Cantor. The question is not, if the naturals are countable or not. The question is, if the 2th diagonal argument of Cantor is senseful and valid. But the fact, that there is no algorithm which numbers the reals shows nothing more than exact this fact. It doesn't tell us anything about the "number" of the reals, about the cardinality of a postulated set of the reals.
> > A set is , by definition, countable if and only if there is surjective > > mapping from the naturals onto that set in which every member of tha set > > is paired with at least one natural. That esentially requires the > > members of a countable set to be completely listable, with each natural > > determining a position in the list and every member of the set in > > question being listed. > > > > But Cantor's "diagonal" argument has shown that such a listing cannot be > > done with the reals in such a way as to use up every real in the list. > > > > Thus there is no way to "count" the entire set of reals. > > Any and every attempt to count them will necessarily leave some, in fact > > most, of those reals out. > > --