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**1 - 2**of**2**### Proof Theory Since 1960

"... ould unquestionably be formalized. These obstacles led workers in proof theory to modify H.P. in two kinds of ways. The first was to seek reductions of various formal systems S to more constructive systems S 0 . The second was to shift the aims from foundational ones to more mathematical ones. Exa ..."

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ould unquestionably be formalized. These obstacles led workers in proof theory to modify H.P. in two kinds of ways. The first was to seek reductions of various formal systems S to more constructive systems S 0 . The second was to shift the aims from foundational ones to more mathematical ones. Examples of 1 the former move are the reductions of PA to intuitionistic arithmetic HA, and Gentzen's consistency proof of PA by finitary reasoning coupled with quantifier-free transfinite induction up to the ordinal ffl 0 , TI(ffl 0 ), both obtained in the 1930s (cf. MATHEMATICS, FOUNDATIONS OF, p. 208). The second re-direction of proof theory was promoted especially by George Kreisel starting in the early 1950s; he showed how constructive mathematical information could be extracted from non-constructive proofs in number theory. The pursuit of proof theory along the first of these lines has come to be called relati