incompleteness theorem (was: gun control)

[W. Kiernan]

Wojtek Sokolowski wrote:
> 
> At 06:18 PM 5/21/99 -0400, Barkely Rosser wrote:
> >     I just read both an article in _Scientific American_
> > and a book review in _Mathematical Intelligencer_ on
> > a new biography of the mathematical logician, Kurt
> > Godel, he of the famous Incompleteness Theorem.
> > Apparently when he was examined to become a US
> > citizen in the late 1940s (Albert Einstein and Oskar
> > Morgenstern were present to shepherd the extreme
> > eccentric genius through), they had to repress him from
> > making an extended speech about contradictions in the
> > US constitution that he perceived when the examining
> > judge asked him what he thought of that particular document.
> 
> That is somewhat strange.  Isn't it what his incompleteness theorem
> would predict?  That is, it is impossible to construct a logically
> coherent system a la constitional law (i.e. you would find mutually
> contradicting statements or principles that are logically derived
> from the "axioms"...

No, Godel's proof says you can not construct a system which completely
describes ordinary arithmetic with any finite number of axioms; in fact,
if I'm not mistaken, not even with any countable number of axioms. 
Godel's proof does not assert that there will necessarily be
contradictions in any axiomatic system, though of course it is simple
enough to construct a series of axioms which are mutually contradictory;
it just says that any system of axioms will not completely describe the
arithmetic system.  That is, given any axiomatic system, there are true
statements about arithmetic which can not be proven using those axioms. 

Godel's proof does this by a chasing-your-tail technique of mapping
logical proofs onto arithmetic itself; any proof can be rewritten, using
a special notation, as an equation, either true or untrue, in
arithmetic; then whether that numerical equation is true or false is the
axiomatically "unprovable" arithmetic assertion.

Yours WDK - WKiernan at concentric.net