my forty years on his shoulders - Department of Mathematics

16In (Gödel 1931), Gödel only sketches a proof of his second incompletenesstheorem, after proving his first incompleteness theorem in detail. His sketchdepends on the fact that the proof of the first incompleteness theorem, which isconducted in normal semiformal mathematics, can be formalized and provedwithin (systems such as) PA.Gödel promised a part 2 of (Gödel 1931), but this never appeared. There issome difference of opinion as to whether Gödel planned to provide detailedproofs of his second incompleteness theorem in part 2, or whether Gödelplanned to let others carry out the details.In any case, the necessary details were carried out in (Hilbert, Bernays1934,1939), and later in (Feferman 1960), and most recently, in (Boolos 1993).In (Hilbert, Bernays 1934,1939), the so called Hilbert Bernays derivabilityconditions were isolated in connection with a detailed proof of Gödel’s secondincompleteness theorem given in (Hilbert, Bernays 1934,1939). Later, theseconditions were streamlined in (Jerosolow 1973).We take the liberty of presenting our own particularly careful and clearversion of the Hilbert Bernays conditions.Our starting point is the usual language L = predicate calculus withequality, with infinitely many constant, relation, and function symbols. Forspecificity, we will usei) variables x n , n ≥ 1;ii) constant symbols c n , n ≥ 1;iii) relation symbols R n m, n,m ≥ 1;iv) function symbols F n m, n,m ≥ 1;