FOM: Does Mathematics Need New Axioms?

John Steel steel at
Mon May 1 14:40:12 EDT 2000

Dear Harvey,
   One remark on a tangential point you make concerning the suitability of
-Con(ZFC) as an axiom. You say it is unsuitable because it is
"syntactical". But other places you say you can hit any Con statement on
the nose with some combinatorial equivalent. Your argument against
-Con(ZFC) as an axiom would not apply to its equivalent.
   ZFC + V=L + -Con(ZFC) decides all remotely natural questions we know
of, and is very likely consistent. What's wrong with it on pragmatic
grounds? I think you can find the pragmatic assymetry between this theory
and theories which include Con(ZFC) in the speed-up of proofs of delta_0
statements. E.g. Con(ZFC) enables us to conclude that no proof of an
inconsistency from ZFC will be found in the next 20 years. This becomes a
delta_0 statement if you make some simple assumption about the lengths of
proofs which will be found, and this delta_0 statement has a proof in ZFC,
but THAT proof would take more than 20 years to find! So Con(ZFC), if
true, has a pragmatic advantage over -Con(ZFC) which shows up in the realm
of its consequences in the realm of simpler (i.e. delta_0) statements.

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