FOM: Syntactic vs semantic classification of math statements

Fri Apr 17 17:03:09 EDT 1998

N.B. This was originally sent privately to Steve Simpson but he suggested I
post it to give everyone a crack at it.   I'd also be interested in answers to
the same questions posed for "famous theorems" instead of "important problems."
-- Joe Shipman

  Your response to Shoenfield reminded me of a project of mine.  I am trying to
collect as many important open problems of mathematics as I can and classify
them by logical type.  My list so far (excluding logic, fom, and set theory)
includes: 1) P=?PSPACE, P=?NP (pi_0_2)    2) Poincare conjecture (equivalent to
pi_0_2 because simplicial category = topological and differentiable categories
in dim<=3)  3) Riemann Hypothesis (pi_0_1) OR abc conjecture (I believe pi_0_3)
5) Invariant Subspace problem (as commonly stated this is pi_3_2 but it is
equivalent by coding to a pi_1_2 statement).  6) Standard packing of spheres in
dimension 3 is optimal (equivalent to a pi_0_1 statement)  7) White doesn't
lose chess (pi_0_0)  You know all about codings into 2nd-order arithmetic so I'd

like your opinion on the following: 1)Are any important open problems outside of
logic, fom, and set theory not equivalent to statements of 2nd-order arithmetic?
 2) What important problems require more than 2 quantifiers in 2nd-order arith-
metic? More than 3? 3) What important (1st-order) arithmetical problem needs the
most quantifiers? 4) What's the most important unsolved pi_0_0 problem? -- Joe

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