From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: Human intelligence vs. Machine intelligence
Organization: Boise State University
Date: Mon, 12 Oct 1992 21:40:52 GMT

In article <burt.718784794@aupair.cs.athabascau.ca> burt@aupair.cs.athabascau.ca (Burt Voorhees) writes:
>>     I find it interesting tht people who support what is called "strong AI"
>>   are now trying to get out of the Godel argument against this by saying
>>   that Godel's theorems only apply to consistent systems and it is clear
>>   that any machine which matches human capacities would have to be running
>>   on some sort of inconsistent system.
>
>>I do not know what you are talking about. Do you have references
>>to any books or articles that contain such an argument made by a
>>supporter of strong AI?
>
>  No, there were several postings sent to me e-mail by people in response
>to an earlier posting of mine here.  Each said that, well, we get away from
>Godel because we don't really need consistent systems since human
>intelligense is clearly inconsistent anyway.
>  On the other hand, I have a suspicion that fame and fortune awaits a person
>who can stand Godel on his head and develop a mathematics and logic of
>complete but inconsistent systems...
>
>>If you are interested in a refutation of Lucas's G"odel argument,
>>see [eg] R. Kirk's article in Synthese [1].
>
>  But Lucas's argument is not the only one.  Penrose doesn't really
>follow his Godel argument to it's obvious conclusion.  Namely, we have
>a formal system.  Okay, it has a Godel proposition which we know is trud,
>but which can't be proved within the system.  We generate a bigger formal
>system which can prove this proposition.  Etc.  This gives an infinite
>regress with the structure, mathematically, of an inverse limit system.
>Using the inverse limit theorem we know that such a limit exists but
>by the construction it must have different properties than any other
>forml system in the sequence.  E.g., by construction it must be complete,
>hence inconsistent.  The problem for machine intelligence is that it is
>not possible to reach this limit in a finite number of steps (shades
>of Zeno).  That is, at some point there must be an (inductive) jump
>and some people (e.g., Searle and Penrose) doubt that this can be a
>matter of simply running an algorithmic program.

This is wrong.  The limiting system will not be complete, and will
have its own Godel sentence (if it is formalizable), etc.  If one
carries this out to a sufficiently large infinite ordinal, you will
get a system which cannot be formalized, and so is allowed to be (but
will not necessarily be) complete (not all incompletenesses are
results of the Godel phenomenon).  No inconsistency should be expected
to arise, in any case, if the original system was consistent.

>
>>oz
>>---
>>[1] Roland Kirk
>>    Mental Machinery and Godel
>>    Synthese, 66, 437-452. 1986
>>---
>>It is harder to get lost in an imagined forest than in a real one,
>>for the former assists the thinker furtively. - GOLEM lect. XLIII


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu
Newsgroups: comp.ai,news.answers