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From: olaf@cwi.nl (Olaf Weber)
Subject: Re: Penrose and human mathematical capabilities
In-Reply-To: tim@maths.tcd.ie's message of 12 Jul 1995 04:11:10 +0100
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Date: Wed, 12 Jul 1995 09:40:42 GMT
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Xref: glinda.oz.cs.cmu.edu comp.ai.philosophy:29990 sci.logic:12236

In article <3tvege$gm@bell.maths.tcd.ie>, tim@maths.tcd.ie (Timothy Murphy) writes:
> constab@unixg.ubc.ca (Adam Constabaris) writes:

>> TMs have Godelian limits -- granted.  But where's the proof that
>> humans have transcended the limits of *all* TMs?

> They would only have to "transcend the limits" of one TM, namely a
> universal machine which can emulate all TMs.

This is wrong.  One universal TM could be programmed to prove the
Gdel sentence of a different universal TM.  (And be emulated by the
TM it is proving the Gdel sentence of.)

A proof that humans have no Gdelian limit must imply that they
transcend the limits of all universal TMs, and thus of all TMs.

So far, the closest that anyone has come to such a proof is to state
that humans _seem_ to able to do so.  Something more convincing would
be appreciated.

-- Olaf Weber
