On 15/06/2010 2:13 PM, |-|ercules wrote: > Consider the list of increasing lengths of finite prefixes of pi > > 3 > 31 > 314 > 3141 > .... > > Everyone agrees that: > this list contains every digit of pi (1) > > as pi is an infinite digit sequence, this means > > this list contains every digit of an infinite digit sequence (2) > > similarly, as computable digit sequences contain increasing lengths of > ALL possible finite prefixes > > the list of computable reals contain every digit of ALL possible > infinite sequences (3)
The discussion on permutations of lists of computable reals to construct diagonals showed that it is possible to construct diagonals that have any finite prefix. However, you were quick to point out that as the computable reals contain numbers like 0.111111...., diagonals not containing a 1 are impossible.
That is to say, there is a set of reals which contains all possible finite prefixes, but which does not contain all possible infinite sequences.
Since that contradicts the reasoning used to conclude your proposition (3), you'd have to prove it some other way.