On 18 Aug 2011, at 03:52, Terren Suydam wrote:
Given a machine's inability to prove its own consistency,
OK. That is Gödel's second incompleteness theorem.
and how this
result gives rise to the many logical distinctions that map to the
hypostases (per Plotinus) as you've written, then I wonder what you
would say to this: if a machine is universal, surely it can run a
program that implements a (higher order or more-encompassing) logical
machine that *can* prove the machine's consistency. If so, can't it
use that result to prove its own consistency?
Good question. The answer is no, of course. The machine would prove
its own consistency, and becomes inconsistent (by Gödel's second
RA(*) cannot prove its own consistency (no consistent machine can do).
PA (a much stronger LUM than the UM RA) can prove the consistency of RA.
RA can implement and run PA. Or, RA can emulate PA. But why would RA
believes in PA's axiom and theorem?
That would be an error à-la Searle's chinese room. With enough time
and patience I can emulate the brain of a chinese guy, but this would
not entail that I understand chinese.
I can emulate the brain of Einstein, but that does not mean I agree or
even understand what Einstein is saying.
In math the key here is that computability is absolute and proof is
relative. Searle makes a confusion of level.
From the fact that PA can emulate ZF doing a proof of the consistency
of PA, PA cannot deduce its own consistency, because there is no
reason for PA having the slight understanding of ZF argument.
All right? This is very important, as you guess. The Universal
dovetailer is more or less equivalent, with respect to emulability,
to RA, and it emulates all LUMs, but is not a LUM itself.
(*) Acronym explained:
RA = Robinson Arithmetic
PA = Peano Arithmetic
ZF = Zermelo-Fraenkel Set Theory.
UM = Universal machine (like RA, PA and ZF). I identify machine and
theory (technically this can be done properly thanks to a theorem due
LUM = Löbian Universal Machine (like PA and ZF)
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