[FOM] Intuitionists and excluded-middle
Lew Gordeew
legor at gmx.de
Sat Oct 22 04:56:45 EDT 2005
Jesse Alama wrote:
> Why should we assume that every irrational number
> between 0 and 1 (or any irrational number for that matter), regarded
> as a choice sequence, has such a block of 9's?
We should not assume anything, as this block might be empty (by definition n
>= 0). Decimal expansions are not really necessary, either. It would suffice
to take any pair of rationals r, s such that
r < a < s < b and s-r < b-a
and then let c:= a+s-r. This obviously yields the result. The proof can be
formalized in Constructive Analysis with Markov's principle, since x < u and
u < x are both decidable for any rational x and irrational u. Ergo LEM is
not required.
Regards,
LG
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