[FOM] Intuitionists and excluded-middle
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Thu Oct 20 14:25:25 EDT 2005
On Wed, 19 Oct 2005, Jesse Alama wrote:
> 2. This proof might not go through for all representations of real
> numbers, especially an important one in this connection, namely
> representation by choice sequences. You conclude from the
> assumption that a and b are irrational numbers between 0 and 1 that
> we can represent them as
>
> a = (0.)x_1,...,x_k,y,y_1,...,y_n,z,z_1,z_2,...
> b = (0.)x_1,...,x_k,u,u_1,u_2,...
>
> where the sequence of y_i's is the first (and longest) block of 9's in
> the decimal expansion of a after which a and b "disagree" in their
> decimal expansions. 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?
First, a and b differ in their expansions from the (k+1)-th place (since
y<u). Secondly, Lew Gordeew explicitly allowed the possibility that n
might be 0 (hence that there be no 9s).
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