From: Ben Goertzel (ben@goertzel.org)
Date: Sat Oct 09 2004 - 06:03:48 MDT
Eliezer wrote:
> It's quite easy to prove that his premise 2 is false. You cannot
> correctly
> solve the problem "Will Ben Goertzel solve this problem by choosing the
> answer 'no'?" Feel free to use an appropriate diagonalization
> lemma if you
> object to the self-reference.
Eliezer, what I object to is your using "Ben Goertzel" in your problem
definition without fully formalizing the concept of "Ben Goertzel."
If you try to formalize "Ben Goertzel" using computation theory, then you're
already assuming the thing Penrose is trying to prove -- so the most you can
do is give a Penrosean "reduction ad absurdum" of the assumption that "Ben
Goertzel" can be formalized using computation theory.
Penrose would say that the correct formalization of "Ben Goertzel" involves
some unknown physics, and plausibly involves uncomputably large sets of
propositions...
Diagonalization is odd here here because we don't know anything about the
cardinality of the set of possible statements, since we don't know the
cardinality of the set of possible beings such as "Ben Goertzel", who are
postulated to live in some huge space of uncomputably complex entities.
Penrose is free to define Ben Goertzel as some kind of quasi-measure defined
over the space of all possible properties, including uncomputable ones. He
runs up against various logical paradoxes here but then he can defer their
resolution to future mathematics ;-)
-- Ben G
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