Re: No Unique Initial Segment And No Characteristic Expansion

From: Daryl McCullough (daryl_at_atc-nycorp.com)
Date: 12/10/04 

Date: 9 Dec 2004 20:24:34 -0800

|-|erc says...

>>> I'm amazed grown men believe in this, DMC's diagonalizer is never
>>> unique, not after 1 anti-flip, not after 10 anti-flips,

Right. It's only the *entire* infinite sequence that is unique.
No finite approximation is guaranteed to be unique.

Let d be the diagonalizer list, and let l(n) be the nth randomly
generated list. You claim that d has no unique
initial segment, and that's true (or it could be true). I claim
that d is not on the list, and that's *also* true. They are not
negations of each other, so it is possible for them both to be
true at once.

If we let Q(l1, l2, k) be the statement
"List l1 agrees with list l2 in at least its first k entries". Then
let A be the formula

   forall k, exists n, Q(l(n),d,k)

let B be the formula

   forall n, exists k, not Q(l(n),d,k)

A and B are not contradictory. They are not the
negations of each other. They can both be true
at the same time. The negation of A is not B but
B':

   exists k, forall n, not Q(l(n),d,k)

which differs from B in the order of its quantifiers. 

--
Daryl McCullough
Ithaca, NY
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