Re: Proper Turing cones are null?

From: Mike Oliver (mike_lists_at_verizon.net)
Date: 02/23/05 

Date: Wed, 23 Feb 2005 07:28:13 -0600

Keith Ramsay wrote:
> Mike Oliver wrote:
> | I *think* the following is true -- if x is a noncomputable
> | real, then the set of y such that x is computable wrt y has
> | measure 0.
>
> The set of y such that the n-th bit of x is the same as the
> 2^n-th bit of y for n>=1 has positive measure.

You had me going for a bit, as this claim kind of sounds
true. But it isn't. For y to satisfy this criterion up
to its (2^k-1)st bit, it has to satisfy k independent constraints,
each probability 1/2 (I guess we're working in Cantor space
with coin-flipping measure). The probability of that
is 1/2^k. As k->infty, the probability goes to zero.

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