On Friday, December 6, 2013 7:17:59 PM UTC+1, Zeit Geist 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. > > > > Wow, your right and I concede. > > > > NOT! > > > > This is so wrong on so many level. > > > > And, isn't worth a response. > > > > ZG
Okay. Let's make some tests to see if and how my algorithm works.
Here is a list of naturals: 1, 2, 3 the construction of the antidiagonal gains d_1 = 2, d_2 = 3, d_3 = 4. Here the list ends and so d_list = 4. And it's true: 4 is not in the list.
Another list 5, 18, 36, 3 so d_1 = 6, d_2 = 6, d_3 = 6, d_4 = 6. Here the list ends. So d_list = 6. In fact, 6 is not in the list.
Now let's have a more sophisticated list: 1, 2, 3, 5, 6, 7, 8, ... so d_1 = 2, d_2 = 3, d_3 = 4, d_4 = 4, d_5 = 4, d_6 = 4, ... . The antidiagonal doesn't change over the rest of the list, so d_list = 4. And correct, 4 is not in the list.
Now some would argue that I can't give the antidiagonal of the list 1, 2, 3, 4, 5, ...
So what. There are a lot of lists of reals at which nobody can give the exact antidiagonal constructed by the second diagonal argument of Cantor. Maybe some digits. But some digits are nothing since the infinite digits of the residual are absent.
So Cantor's second diagonal argument fails as much as my first diagonal proof of the uncountability of the naturals fails.