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From: ma261aai@sdcc3.ucsd.EDU (Stephen Bloch)
Newsgroups: sci.philosophy.tech
Subject: Re: Godel (short)
Message-ID: <4075@sdcc3.ucsd.EDU>
Date: Fri, 9-Oct-87 01:46:05 EDT
Article-I.D.: sdcc3.4075
Posted: Fri Oct  9 01:46:05 1987
Date-Received: Sun, 11-Oct-87 13:23:37 EDT
References: <2362@sphinx.uchicago.edu> <10368@duke.cs.duke.edu>
Reply-To: ma261aai@sdcc3.ucsd.edu.UUCP (Stephen Bloch)
Organization: University of California, San Diego
Lines: 51

In article <10368@duke.cs.duke.edu> mps@duke.UUCP (Michael P. Smith) writes:
>
>In article <2362@sphinx.uchicago.edu> hin9@sphinx.UUCP (The Reverend w. No Name) writes:
>>
>>     'This Sentence is Unprovable in System S.'
>>
>>     If it's false, it is provable in system S, which is impossible unless
>>S is inconsistent.  Therefore, it is true.  Therefore, it is unprovable
>>in system S.
>
>One reason the proof is so short is that you don't show how the
>provability of the godelian sentence in S would entail that S is
>inconsistent. 

If you accept the semantic definition of inconsistency, i.e. "S is
inconsistent if it has a theorem which on the intended interpretation
is false," it's obvious.  If, however, you want a purely proof-
theoretic definition, i.e. "S is inconsistent if every wff in it is a
theorem," there is indeed some nontrivial content yet to come.

The other reason it's so short is because it contains the word
"this".  Does "This sentence" mean the sentence following the one in
which "this" appears, or the sentence preceding the one in which
"this" appears (for both of which possibilities there are plentiful
examples), or the one the author is thinking about at the moment the
reader reads the word "this", or the sentence the _reader_ was
thinking about at the moment the _author_ first typed the word
"this", or any number of other possible sentences that aren't the
intended one (which, after all, is a VERY unusual way to interpret
the word "this")?  I think to avoid this (literally), the shortest
possible example looks something like

<If you replace all but the first occurrence of <snark> in <If you
replace all but the first occurrence of <snark> in <snark> with
<snark>, the result is unprovable in system S.> in <If you replace
all but the first occurrence of <snark> in <snark> with <snark>, the
result is unprovable in system S.>, the result is unprovable in
system S.>

Such a sentence does not rely on an intuition that we all know what
"this" means, but only on the notions of string sequence and
substitution, which can easily be defined purely formally.  It
doesn't assume that everyone reading it correctly understands that
"this sentence" is a self-reference, but actually CONSTRUCTS the
sentence it's talking about, which just happens to be identical to
itself.
Aside: It disturbs me somewhat that if I wrote the same sentence
using quotation marks " and ' that are undistinguished between left
and right, I couldn't possibly make it both unambiguous and character-
for-character self-referential.  Does this perhaps say something
about the inherent logical superiority of nesting syntax?