my forty years on his shoulders - Department of Mathematics

24directly formalizing the syntax of T, including a direct formalization of theconsistency of T, then T’ does not prove the consistency of T (so expressed).We can recover the usual second incompleteness theorem for T from theabove direct second incompleteness, by proving that there is an interpretation ofT’ in T. This was also done in (Quine, 1940, 1951, Chapter 7).Thus under this view of second incompleteness, one does not view Con(T)as a sentence in the language of T, but instead as a sentence in the language of anextension T’ of T. Con(T) only becomes a sentence in the language of T throughan interpretation (in the sense of Tarski) of T’ in T. There are many suchinterpretations, all of which are ad hoc. This view would then eliminate ad hocfeatures in the formulation of second incompleteness, while preserving thefoundational implications.5. LENGTHS OF PROOFS.In (Gödel 1936), Gödel discusses a result which, in modern terminology,asserts the following. Let RTT be Russell’s simple theory of types with the axiomof infinity. Let RTT n be the fragment of RTT using only the first n types. Let f:N→ N be a recursive function. For each n ≥ 0 there are infinitely many sentences ϕsuch thatf(n) < mwhere n is the least Gödel number of a proof of ϕ in RTT n+1 and m is the leastGödel number of a proof of ϕ in RTT n .Gödel expressed the result in terms of lengths of proofs rather than Gödelnumbers or total number of symbols. Gödel did not publish any proofs of thisresult or results of a similar nature. As can be surmised from the Introductory