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From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Review of Shadows of the Mind
Message-ID: <1995Apr1.190000.6415@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Sat, 1 Apr 1995 19:00:00 GMT
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rickert@cs.niu.edu (Neil Rickert) writes:

> Goedel's theorem does not show that there must be a mathematical
> truth which is undecideable.  All it shows is that there must be
> undecideable propositions.  In order to claim that the Goedel
> proposition is actually true, you already have to assume something
> like Platonism.  So all you have shown is that if you assume
> Platonism, then on the basis of that assumption and Goedel's theorem,
> you can prove Platonism.

Neil,

Whether or not Goedel's theorem proves Platonism, you don't have to
assume Platonism to know that the Goedel statement for Peano
Arithmetic is true (if PA is consistent). As a matter of fact, the
statement "If PA is consistent, then G", where G is the Goedel
statement for PA, is a theorem of PA (where consistency is interpreted
as the nonexistence as a number coding a proof of false).

> As far as I am concerned, mathematical truth consists of provability
> within a formal system.  All mathematical truths are implicitely of
> the form: if {contextually assumed axiom system} then {proposition}.
> Note that in this version of truth, the claim that the Goedel
> proposition is true, is an incorrect claim.  You can say that it is
> true in the axiom system that Goedel used (the Meta-system), but you
> cannot say that it is true independent of the axiom system.

You are misusing the phrase "mathematical truth". What you are calling
a "mathematical truth", everyone except you would call a "theorem".
Truth is not a property of a formal system alone, but of a formal
system together with an *interpretation* of the formulas.

However, by Goedel's Completeness Theorem (as opposed to his
*IN*completeness theorem), every consistent theory *can* be given an
interpretation so that all its statements are true. But usually when
people ask whether something is true or not, they have an intended
interpretation in mind.

Daryl McCullough
ORA Corp.
Ithaca, NY
