Applications of cut-free infinitary derivations to generalized recursion theory

Proof Theory

“…In case of a well-ordering, however, there is a strengthening of Gentzen's theorem due to Beckmann [8] which we are going to prove next. In case of a well-ordering, however, there is a strengthening of Gentzen's theorem due to Beckmann [8] which we are going to prove next.…”

“…Nevertheless, the extensions needed seemed to be so tiny that Bernays (loc. 8 So he succeeded in giving a consistency proof in which the "nonfinitist" means are concentrated in a single principle, an induction along a primitive-recursively definable well-ordering of the natural numbers of transfinite order type ε 0 . ) argued that Gentzen's proof meets the basic requirements of Hilbert's finitist standpoint.…”

“…[8]). Therefore we can extend Lemma 6.6.8 and Theorem 6.6.9 to binary relations which are Σ 1 1 -definable.…”

“…8 Secondly, we have to secure that the cut-elimination operations, as given by the Reduction Lemma (Lemma 7.3.12) and Lemma 7.3.13, are primitive recursive, hence executable in NT 0 . Since this leads us away from the main topic of this book, we will be quite sketchy.…”

“…8 The abstract background is the hyperarithmetical quantifier theorem which states that every Π 1 1 -relation is uniformly equivalent to a relation which is Σ -definable on L ω CK 1 . This requires that L ω CK 1 is definable in these theories.…”