Newsgroups: comp.ai.philosophy,sci.logic
Path: cantaloupe.srv.cs.cmu.edu!nntp.club.cc.cmu.edu!miner.usbm.gov!rsg1.er.usgs.gov!stc06.CTD.ORNL.GOV!fnnews.fnal.gov!mp.cs.niu.edu!vixen.cso.uiuc.edu!howland.reston.ans.net!pipex!uknet!festival!edcogsci!jeff
From: jeff@aiai.ed.ac.uk (Jeff Dalton)
Subject: Re: Penrose's new book
Message-ID: <CyruDB.K5K@cogsci.ed.ac.uk>
Sender: usenet@cogsci.ed.ac.uk (C News Software)
Nntp-Posting-Host: bute-alter.aiai.ed.ac.uk
Organization: AIAI, University of Edinburgh, Scotland
References: <1994Oct24.141552.20925@oracorp.com>
Date: Sat, 5 Nov 1994 01:48:47 GMT
Lines: 43
Xref: glinda.oz.cs.cmu.edu comp.ai.philosophy:21686 sci.logic:8825

In article <1994Oct24.141552.20925@oracorp.com> daryl@oracorp.com (Daryl McCullough) writes:
>zeleny@oak.math.ucla.edu (Michael Zeleny) writes:
>
>>Penrose imputes a certain closure property to the theory of human
>>cognitive performance.
>
>Yes, an inconsistent closure property. Basically, Penrose assumes that
>human intuition can show us that human intuition is infallible. But
>the belief that one is infallible, together with sufficient closure
>properties of ones beliefs, implies that one is in fact inconsistent.
>
>>(The last cited characteristic rules out Turing's conjecture of human
>>inconsistency, since performance relativized to time is consistent by
>>definition.)
>
>It may be true that no person can hold two contradictory beliefs at
>the same time. But Penrose was not talking about the collection of
>statements believed to be true at one time---he was claiming that the
>collection of statements that will ever be held to be "unassailably
>true" is a consistent, noncomputable set. His claim is certainly not
>true by definition.
>
>>The property in question involves the ability to judge the consistency
>>of an arbitrarily complex formal system.  (Compare the ascent of
>>reflection principles in some transfinite progression of ordinal
>>logic.)  It is highly implausible that any finite increase in
>>complexity will a priori rule out the possibility of making a correct
>>judgment in this matter.
>
>No, it is not highly implausible, it is highly likely. Using
>Penrose-style arguments, one can go from PA to PA + con(PA) to PA +
>con(PA + con(PA)), etc. (where "con" is the consistency predicate on
>r.e. theories). As you point out, one can in fact systematize this
>progression as follows: Let r be any computable relation on integers,
>and define the infinite sequence of theories T_j:

> [...]

Did Michael Zeleny reply to this, BTW?

If so, I haven't seen it, and I'd like to see what he says.

-- jeff
