FOM: June 25 - July 31, 1999

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FOM: De Branges claims to have shown measurable cardinals inconsistent

Louis De Branges of Purdue University has claimed on his web page

to have shown that no nontrivial countably additive measure exists on
all subsets of the continuum (which by Solovay implies that there are no
measurable cardinals).  His manuscript is available at

, along with manuscripts containing purported proofs of the invariant
subspace conjecture and the Riemann Zeta Hypothesis and an
autobiographical "apology" which explains the mathematical motivation
for his Riemann Hypothesis proof and reveals that he proved the
Bieberbach conjecture so that he could get funding to work on the
Riemann Hypothesis.

The "Measure Problem" manuscript begins with a generalization of Stone's
proof of the Hahn-Banach theorem to apply to  "hyperconvex sets".  The
last 5 pages apply this representation theorem to obtain the result on
measures.  De Branges uses nonstandard terminology for the
set-theoretical part of the paper, which makes it more difficult to
follow.  It would be easy to dismiss De Branges as a crank because of
this, but he has earned the right to a hearing because the early
dismissals of his work on the Bieberbach Conjecture turned out to be
wrong.  As far as I can tell, his Riemann Hypothesis and Invariant
Subspace manuscripts are more conventional in their terminology.

Although a refutation of measurable cardinals would be very surprising,
and I strongly doubt De Branges has accomplished this, other eminent
mathematicians who are experts in the field have admitted to trying to
refute measurable cardinals, and the example of Cohen shows that it is
not necessary to be a specialist in set theory to make a breakthrough in
the field.

-- Joe Shipman

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