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Article 1394 of comp.ai.philosophy:
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>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: comp.ai.philosophy
Subject: Re: Chinese Room Variant
Message-ID: <1991Nov19.010301.5592@husc3.harvard.edu>
Date: 19 Nov 91 06:02:59 GMT
References: <1991Nov18.114641.964@CSD-NewsHost.Stanford.EDU> 
 <1991Nov18.094715.5563@husc3.harvard.edu> <1991Nov19.001943.9064@CSD-NewsHost.Stanford.EDU>
Organization: Dept. of Math, Harvard Univ.
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Nntp-Posting-Host: zariski.harvard.edu

In article <1991Nov19.001943.9064@CSD-NewsHost.Stanford.EDU> 
costello@CS.Stanford.EDU (Tom Costello) writes:

>In article <1991Nov18.094715.5563@husc3.harvard.edu>, 
>zeleny@osgood.harvard.edu (Mikhail Zeleny) writes:

>|> In article <1991Nov18.114641.964@CSD-NewsHost.Stanford.EDU> 
>|> costello@CS.Stanford.EDU (Tom Costello) writes:

>|> >In article <1991Nov17.163705.5540@husc3.harvard.edu>, 
>|> >zeleny@brauer.harvard.edu (Mikhail Zeleny) writes:

>|> >|> In article <1991Nov14.163630.20597@spss.com> 
>|> >|> markrose@spss.com (Mark Rosenfelder) writes:

>|> >|> >In article <1991Nov12.131428.4850@osceola.cs.ucf.edu> 
>|> >|> clarke@next1 (Thomas Clarke) writes:

TC:
>|> >|> >>That is, the bare rules of computation plus any finite set of additional 
>|> >|> >>usage/correspondence rules are not sufficient for an understanding of number. 

MR:
>|> >|> >What is sufficient, then?  An infinite set of rules?  Or, if something else
>|> >|> >entirely is needed, what is it?

MZ:
>|> >|> In effect, on the syntactical level, infinitely many rules are required in
>|> >|> order to characterize the standard model of the natural numbers.  This is a
>|> >|> direct consequence of G\"odel's Second Incompleteness Theorem.

TCos:
>|> >While it is very flattering to see Mikhail repeat something that I explained
>|> >to him just recently on sci.philosophy.tech,

MZ:
>|> yeah, sure.

Um, Tom, have you figured out a way to give a finitist definition of
finitude yet?
 
TCos:
>|> >                                             I should point out that the
>|> >theorem is Godel first incompleteness theorem.   Also, infinitely many
>|> >rules
>|> >is not a correct characterisation.  It should be, and I quote from
>|> >Godel Collected Work's, editor Fefermann.

TCos:
>The following is a quotation, I'll put in quotation marks so 
>it's even more obvious. 
> "
>|> >1. The class of axioms and rules of inference(that is, the relation "immediate
>|> >consequence") are recursively definable ( as soon as we replace the primitive signs in some way by natural numbers).
>|> >2. Every recursive relation is definable in the system"

MZ:
>|> Had your grade school teachers been more attentive to your development, you
>|> might have learned to read; then you would be able to distinguish a
>|> consequence from a quotation (not that you really deliver on your promise
>|> to present the latter).

TCos:
>The above is a quotation.  It concerns the conditions necessary for 
>the application of Godel's First Incompleteness Theorem.

I, on the other hand, was talking about a consequence.

MZ:
>|>  Had you taken the time to read the Feferman
>|> edition you are citing, you might have discovered that G\"odel never
>|> published a proof of the Second Incompleteness Theorem.
    ^^^^^^^^^^^^^^^^^
TCos:
>Ok, this is false, here is a claim that he did, in Godel's words;
>"Satz XI.  Sei k eine beliebige rekursive widerspruchsfrie Klasse
>von FORMELEN, dann gilt.  Die SATZFORMEL, welche besagt, dass k
> wider-spruchsfrei ist, ist nicht k-BEWEISBAR; insbesondere ist die
> Widerspruchs-freiheit von P in P unbweiesbar, vorausgesetzt, dass P
> widerspruchsfrei ist ( im entgegengesetzten Fall ist naturlich jede
> Aussage beweisbar).
> Der Beweis ist ( in Umrissen skizziert) der folgende: .. "
>And in translation;
>Theorem 11.  Let k be any recursive consistent class of FORMULAS; then the
>SENTENTIAL FORMULA stating that k is consistent is not k-PROVABLE; in
>particuylar the consistency of P is not provable in P, provided P is
>consistent ( in the opposite case, of course, every proposition is provable
>[in P]).
>The proof (briefly outlined) is as follows:...
            ^^^^^^^^^^^^^^^^
Read my lips: the complete proof has never been published.

TCos:
>All of the above is taken from the aforementioned volume.

Read the paper to the end, then read Kleene's comments on p.137.  

MZ:
>|>  All interested
>|> parties are referred to Bell and Machover, or Smorynski's article in the
>|> "Handbook of Mathematical Logic".

TCos:
>At least this interested party will read the original.  Godel second 
>incompleteness theorem, is the final theorem, Theorem XI of section 4
>of Godel 1931.  Misha, read the originals, sometimes commentators
>get it wrong.

Actually, in this case the commentators (i.e. Hilbert & Bernays, L\"ob, and
Smory\'nski) are the only ones to furnish the proof.

TCos:
>|> >The first part is the condition that the system must be weaker than for
>|> >the result to apply, the second, a condition that the system must be as 
>|> >strong as.

MZ:
>|> Um, Tom, do you think you could try writing complete English sentences the
>|> next time around?  You know, the ones you might have found in-between the
>|> algebraic formulae over the two decades of your study of the subject...

TCos:
>"The first part" subject, "is" verb, "the condition", object, "..."
>a clause modifying the object, "the second" secondary subject, "a condition"
>secondary object, ".." a modifier of the secondary object;
>These are the parts of speech that the various parts of the sentence
>are.  It is perfectly grammatical, though some would find ending a
>sentence with "as", behaviour up with which they would not put.

I was referring to the dangling comparatives.  Never mind; just fix your
news reader.  Got that standard model of arithmetic under control yet?

>Tom


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