Path: utzoo!utgpu!news-server.csri.toronto.edu!bonnie.concordia.ca!uunet!wuarchive!udel!haven.umd.edu!socrates.umd.edu!socrates!rockwell From: rockwell@socrates.umd.edu (Raul Rockwell) Newsgroups: comp.theory Subject: Re: The Ulam Machine (was: Re: Paradox of the non-recursive computer) Message-ID: Date: 12 May 91 20:29:38 GMT References: <11943@uwm.edu> <12089@uwm.edu> Sender: rockwell@socrates.umd.edu (Raul Rockwell) Organization: Traveller Lines: 25 In-Reply-To: markh@csd4.csd.uwm.edu's message of 12 May 91 18: 18:40 GMT Mark William Hopkins writes: 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") Where he defined T0, T1, ... as: 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. Each set is incremented from the previous in the following way: 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. However, he fails to address the issue of determining "if statement An is consistent with Tn + axioms T". It looks like he is extending a theorem by adding some some compatible theorem as an axiom. Since substitution allows one to trivially obtain a compatible theorem, computability is satisfied (as is turing computability). Am I just being dense? Raul Rockwell