[FOM] Question on Equivalence of Theorems

[Stephen G Simpson]

Harvey Friedman writes:
 > You should become familiar with Reverse Mathematics, if you aren't 
 > already, a highly developed part of f.o.m. Steve Simpson wrote the 
 > definitive book on it, published by Springer in 1999.

The first chapter of my book is on-line at
http://www.math.psu.edu/simpson/sosoa/.  The book quickly went out of
print, but there are plans to reprint as soon as I can do a little
editorial work on it.

I like Reverse Mathematics a lot, but I don't want to claim too much
for it.  It doesn't provide an analysis of "equivalence of theorems"
in quite the sense that Robin Adams was referring to.  Reverse
Mathematics exposes an obviously important and interesting
classification of mathematical theorems up to logical equivalence, but
it doesn't take account of the complexity of the equivalence proofs.
Robin Adams asked about "simple" equivalence proofs.

By the way, Ruediger Thiele's article about the so-called "Hilbert's
24th Problem" (which Hilbert omitted from the famous problem list) has
now appeared in the January issue of the American Mathematical
Monthly.

In his notes on the "24th Problem", Hilbert points to metamathematical
issues concerning the strength of existence axioms in mathematics, the
simplest proof of a theorem, etc etc.  When I read Hilbert's comments,
it seemed obvious that he was anticipating Reverse Mathematics and
related programs.  But maybe that was only my prejudice.

-- Steve