From: Eric Jablow <ejablow at cox.net>
Subject: [WSFA] Sad math news
Date: Tue, 27 Mar 2007 23:13:43 -0400
To: WSFA members <WSFAlist at KeithLynch.net>
Reply-To: WSFA members <WSFAlist at KeithLynch.net>

Paul Cohen, winner of the 1966 Fields Medal, died last Friday.  His
primary work field was mathematical logic and set theory, and his
most famous result was the counterpart to Kurt G=F6del's most
famous result.  He was 72.

As most of you know, Kurt G=F6del had proved in the 1930s his
famous incompleteness theorems:

1. In any sufficiently-rich consistent system of logic, there are
statements that are true in every valid model, but which cannot
be proved.  Specifically, the statement that the axioms are consistent
is an example.

2. In any sufficiently-rich consistent system of logic, there are
statements that are undecidable from the axioms.  They can
neither be proved nor disproved.  They are called 'independent'.

G=F6del showed that the Axiom of Choice [AC] and the
Generalized Continuum Hypothesis [GCH] could not be
proved from the standard theory of sets, but he didn't
show they were independent.  Paul Cohen, using his
new theory of 'forcing', proved in 1962 that they were
independent.  In other words, G=F6del showed that adding
AC or GCH or both to set theory cannot cause any
contradictions.  Cohen showed that adding their
negations also cannot cause contradictions.  In much
the same way that the Klein or Poincar=E9 models of
Lobachevskian geometry show that one cannot
prove the parallel postulate from the other postulates
of geometry, Cohen's models of ZF and not AC or not
GCH show that ZF doesn't give proofs of AC or GCH.

After this, other more understandable consequences of AC
are clearly independent of the usual axioms.  AC implies the
existence of the Banach-Tarski paradox, that a ball can be
written as the disjoint union of finitely-many sets that can be
moved around to form a bigger ball.  So, BT is independent,
and so on.

Respectfully,
Eric Jablow