FOM: Answer to Pratt -- What if ZF were inconsistent?
JSHIPMAN@bloomberg.net
JSHIPMAN at bloomberg.net
Thu Nov 20 14:14:27 EST 1997
Although Godel's independence results certainly "shook the foundations" they
did not mean that any mathematicians not actually working on Hilbert's program
needed to do anything different. Today the mathematicians who would need to do
something different if an inconsistency were discovered in ZF include all set
theorists, of whom there are certainly more than there were mathematicians
trying to complete Hilbert's program. The entire field of set theory as
currently practiced would collapse. As for "ordinary" mathematicians, you are
right that most would not have a serious adjustment to make but some would need
to go back and rewrite their textbooks and papers to clarify their uses of
Replacement; this is still more than they had to do in 1931. Godel's result is
more of an earthquake outside of math than an inconsistency in ZF would be, at
least philosophically--sociologically I am not so sure. Previous inconsistencies
in mathematical systems were discovered early, so only individuals like Frege
were burned and not whole subfields! Is there any precedent (at least since
the Pythagoreans drowned the discoverer of irrationals) for this?--Joe Shipman
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