An uncomputable number can not be expressed as a finite sequence of digits in any computable base. So Chaitin's constant must consist of an infinite number of digits regardless of what base you choose, so long as the base is computable.
So "God" or an oracle can never actually produce Chaitin's constant in any finite representation, all an oracle can do is behave as a function where you give it an integer N, and it returns the N'th digit of Chaitin's constant.