FOM: Friedman's independence results, an epochal f.o.m. advance

[Stephen G Simpson]

I would like to call attention to Harvey Friedman's posting

       FOM: 12:Finite trees/large cardinals

of 11 Mar 1998 11:36:36.  It represents tremendously important
progress in f.o.m.  One of the key issues in f.o.m. is whether new
axioms are needed.  Opinion is divided, as witness the earlier
discussion of Sol Feferman's paper "Does mathematics need new axioms?"
here on the FOM list.  The background here is G"odel's incompleteness
theorem, which can be interpreted as showing that new axioms are
always needed, simply in order to prove the consistency of the current
axioms.  But a key question that remains is the impact on mathematical
practice.  Harvey's work tends to bring the incompleteness phenomenon
into the realm of core mathematics.  In the research cited above,
Harvey shows that large cardinal axioms (subtle cardinals, etc) are
needed in order to decide some very natural combinatorial propositions
about inserting new nodes into finite labeled trees.  This represents
a new level of achievement in this important direction of
f.o.m. research.

-- Steve