> It is necessary for Cantor's proof. Unless every digit exists on the > diagonal, the diagonal number is undefined in an undefined list. But > that is Cantor's claim: forall n : a_nn =/= d_n.
It exists "as formula" , whether or not you know the formula . Let's define an irrational number , such that it's k'th digit is the k'th prime number mod 10 .
i = 0.235713793......
What do you gain by writing the digits? Nothing . There is nothing that can be done writing the digits that can't be done with the formula. The formula tells you 'more' than how to write down the digits . You'd die of thirst next to a lake because you don't know how to drink water that's not from a 'finite' bottle .
> Yes, but there are only countably many formulas.
Formulas as "strings of characters"/syntax are defined relative to a language . There is no 'completed language' that proves that formulas are countable . A language cannot count over its own formulas:
> The claim that all rationals can be enumerated is already nonsense > (since for every enumerated q_n there are infinitely many not > enumerated - you can enumerate every rational but not all). The claim > that all reals cannot be enumerated is correct (infinity is simply > infinite), but interpreted on the basis of the first claim it is > nonsense too.
> You demand a contradiction of astrology by an astrological argument? I > conradict nonsense by showing that it is nonsense. > > Regards, WM
I (and many others ) demand a contradiction in mathematics by a mathematical argument . Because nobody here understands astrological arguments . You can dabble in you 'astrology' however you want, just don't confuse what you're doing with mathematics . It's a waste of time both for you , and legitimate mathematicians .