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From: rockwell@socrates.umd.edu (Raul Rockwell)
Newsgroups: comp.theory
Subject: Re: Paradox of the non-recursive computer
Message-ID: <ROCKWELL.91May8082000@socrates.umd.edu>
Date: 8 May 91 13:20:00 GMT
References: <11943@uwm.edu>
Sender: rockwell@socrates.umd.edu (Raul Rockwell)
Organization: Traveller
Lines: 18
In-Reply-To: markh@csd4.csd.uwm.edu's message of 8 May 91 08: 01:54 GMT

Mark William Hopkins:
   You can enumerate all the statements of a formal logic system, A0,
   A1, A2, ...  Consider a family of sets T0, T1, T2, ..., built on an
   axiom base T:
     T(n+1) = T(n) + [An] if statement An is consistent with Tn + axioms T.
     T(n+1) = T(n) + [not An] if statement An is inconsistent with Tn + T.
   Since each set is a finite set of statements, it's computeable.

   Consider an abstract machine that correctly emulates T0, T1, T2,
   and so on.  It exists, according to the Axiom of Choice, but is
   more powerful than a Turing Machine (and thus "non-computeable")

er... while it may exist for formal theories, I don't see how that
makes it more powerful than a TM.

In other words, what's the paradox??

Raul Rockwell