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From: fulk@sunybcs.UUCP (Mark Fulk)
Newsgroups: net.ai
Subject: Re: undecidability of undecidability
Message-ID: <1083@sunybcs.UUCP>
Date: Tue, 21-Feb-84 16:22:11 EST
Article-I.D.: sunybcs.1083
Posted: Tue Feb 21 16:22:11 1984
Date-Received: Thu, 23-Feb-84 04:28:57 EST
References: <592@sbcs.UUCP>
Organization: SUNY/Buffalo Computer Science
Lines: 38

The problem with the "undecidability of undecidability" is not
that undecidable problems must have been proved so; it is perfectly
possible for a problem to be undecidable and for us to never know
about it.  Goldbach's conjecture is decidable; either it is true
of the natural numbers or it is not.  If it is, the program that
answers "yes" to any question decides the question "Is Goldbach's
conjecture true"; similarly another program suffices if Goldbach's
conjecture is false.  Goldbach's conjecture may, however, be
unprovable in a particular axiomatization of the natural numbers;
there is always a larger axiomatization which implies it or its
negative.

If Goldbach's conjecture is false, of course, its falsity is provable
from the axioms of Peano arithmetic.  This is true of any proposition
of the form (for all x)[decidable property of x].  Being the sum of
two primes is of course a decidable proposition, since the primes are
constrained to be less than the given number.

The following predicates are undecidable:
"Is x provable in Peano arithmetic"
"Is x a decidable predicate on the natural numbers".
Note that propositions (which are always either true or false)
are always decidable; although perhaps not provable in given axiom
systems.  It is predicates (which have a parameter) which can be undecidable
in the sense that there may not be a program to determine from x,
if R(x) is true.

-- 
Mark Fulk
ARPA and CSNET:	fulk.buffalo@rand-relay
UUCP:		...!seismo!rochester!rocksvax!sunybcs!fulk

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