my forty years on his shoulders - Department of Mathematics

29Harrington 1977), and are proved just beyond PA. We discovered manyexamples in connection with theorems of J.B. Kruskal (Kruskal 1960), andRobertson, Seymour (Roberton, Seymour 1985, 2004), which are far stronger,with no predicative proofs. See (Friedman 2002b).None of these three references discusses the connection with sizes ofproofs. This connection is discussed in (Smith 1985 132-135), and in theunpublished abstracts (Friedman 2006a-g) from the FOM Archives. 3The basic idea is this. There are now a number of mathematically naturalΠ 0 2 sentences (∀n)(∃m)(R(n,m)) which are provably equivalent to the 1-consistency of various systems T. One normally gets, as a consequence, that theSkolem function m of n grows very fast, asymptotically, so that it dominates theprovably recursive functions of T.However, we have observed that in many cases, one can essentiallyremove the asymptotics. I.e., in many cases, we have verified that we can fix n tobe very small (numbers like 3 or 9 or 15), and consider the resulting Σ 0 1 sentence(∃m)(R(n,m)). The result is that any proof in T (or certain strong fragments of T)of this Σ 0 1 sentence must have an absurd number of symbols - e.g., an exponentialstack of 100 2’s. Yet if we go a little beyond T, we can prove the full Π 0 2 sentence(∀n)(∃m)(R(n,m)) in a normal size mathematics manuscript, thereby yielding aproof just beyond T of the resulting Σ 0 1 sentence R(n,m) with n fixed to be a small(or remotely reasonable) number. This provides a <strong>my</strong>riad of mathematicalexamples of Gödel’s original length of proof phenomena from (Gödel 1936).6. THE NEGATIVE INTERPRETATION.3 See http://cs.nyu.edu/pipermail/fom/