X-Message-Number: 4139
From: Eli Brandt <>
Subject: G\"odel
Date: Mon, 3 Apr 1995 03:10:37 -0700 (PDT)

> From: 

> It is claimed that in every mathematical system or language one can make a
> true statement corresponding (more or less) to the following:
> 
> (1)               This statement is unprovable.

To be more precise, the statement is not actually self-referential.
It states a property of a particular gargantuan number.  This turns
out to be equivalent to the above.  So what?  Well, this makes it more
difficult to claim that the statement is empty tail-chasing without
actual content.  Smullyan's and Kleene's books should contain the
formal version of the Godel statement (called "G"); I won't try to
reproduce it from memory.

> The mathematician's answer seems to be that such a statement, represented
> symbolically in a particular mathematical or logical language, is not
> provable "within the system"--but by going outside the system, working in a
> larger universe of discourse, we can prove it. But then--they claim--one can
> still make similar statements in that larger language, which again cannot be
> proven except by recourse to a still larger language, etc. But even if that
> were true, 

It is true.  You've summarized the situation accurately.  What the
theorem says is that no formal system as strong as number theory can
be complete.  That is, there will be a well-formed statement which is
undecidable within the system; it can neither be proved nor disproved.
(A "formal system" is a set of axioms and a set of derivation rules
for creating true statements.  To "prove" is to derive from axioms.
To "disprove" is to "prove" the negation of.)

> it does not address the problem of a valid proposition, nor would
> it appear to ungibberish the English version.

The English version is irredeemable gibberish.  "Formal system"
doesn't translate into lay English at all, and gets left out;
translating the Godel statement requires the introduction of
self-reference, which wasn't there to begin with.  Don't attack a
strawman -- read the proof in the original.

A loosely analogous situation can be found in geometry.  The
"absolute" axioms of geometry (the first four) turn out to be
insufficient to prove or disprove the parallel postulate.  The PP is
nonetheless valid and meaningful.  This situation actually makes a
mathematician happy, because it means you can add either PP or its
negation and get a consistent system.  The situation with Zorn's Lemma
and the continuum hypothesis is similar.

Adding the Godel statement G doesn't evade the theorem, by the way; you
can just apply it again to get a new true-but-unprovable statement G'.
Adding ~G is more interesting, ultimately resulting in a beautiful
area of math called non-standard analysis, which among other things
provides a rigorous justification and methodology for the use of
infinitesimals, which fell out of fashion for lack of rigor.  (You may
ask how we can legitimately add the negation of a "true" statement.
This is left as an exercise for the skeptical reader...)

-- 
   Eli   

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