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From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Penrose's new book
Message-ID: <1994Oct24.141552.20925@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Mon, 24 Oct 1994 14:15:52 GMT
Lines: 61
Xref: glinda.oz.cs.cmu.edu comp.ai.philosophy:21278 sci.logic:8700

zeleny@oak.math.ucla.edu (Michael Zeleny) writes:

>Penrose imputes a certain closure property to the theory of human
>cognitive performance.

Yes, an inconsistent closure property. Basically, Penrose assumes that
human intuition can show us that human intuition is infallible. But
the belief that one is infallible, together with sufficient closure
properties of ones beliefs, implies that one is in fact inconsistent.

>(The last cited characteristic rules out Turing's conjecture of human
>inconsistency, since performance relativized to time is consistent by
>definition.)

It may be true that no person can hold two contradictory beliefs at
the same time. But Penrose was not talking about the collection of
statements believed to be true at one time---he was claiming that the
collection of statements that will ever be held to be "unassailably
true" is a consistent, noncomputable set. His claim is certainly not
true by definition.

>The property in question involves the ability to judge the consistency
>of an arbitrarily complex formal system.  (Compare the ascent of
>reflection principles in some transfinite progression of ordinal
>logic.)  It is highly implausible that any finite increase in
>complexity will a priori rule out the possibility of making a correct
>judgment in this matter.

No, it is not highly implausible, it is highly likely. Using
Penrose-style arguments, one can go from PA to PA + con(PA) to PA +
con(PA + con(PA)), etc. (where "con" is the consistency predicate on
r.e. theories). As you point out, one can in fact systematize this
progression as follows: Let r be any computable relation on integers,
and define the infinite sequence of theories T_j:

    T_k = the set of all statements Phi such that
          Phi is in PA, or for some j such that r(j,k) Phi is in T_j
          or for some j such that r(j,k) Phi = con(T_j)

If r(j,k) is in fact a well-founded relation, then each theory T_k is
consistent. Therefore, the limitation on Penrose' Godelization
technique to get ever more powerful true theories is that any strategy
for coming up with ever larger computable well-founded relations will
eventually peter out. If we stick to what we *know* to be
well-founded, we'll never even do as well as some computable (but not
known to be consistent) theories such as ZFC + "there exists a
measurable cardinal". On the other hand, if we use some very powerful
large cardinal axiom to help us prove that relations are well-founded,
then we have no guarantee that what we are doing is consistent. You
have to give up something; either you take the leap, and give up
certainty, or stay close to the ground, and miss some true facts.
Either way, there is no way to be assured of beating every r.e. theory.

Daryl McCullough
ORA Corp.
Ithaca, NY





