my forty years on his shoulders - Department of Mathematics

34finite type that is based on quantifier free axioms and rules, including a rule ofinduction.The Dialectica interpretation has had several applications in differentdirections. There are applications to programming languages and categorytheory which we will not discuss.To begin with, the Dialectica interpretation can be combined with Godel’snegative interpretation of PA in HA to form an interpretation of PA in Gödel’squantifier free system T.One obvious application, and motivation, is philosophical, and Gödeldiscusses this aspect in both papers, especially the second. The idea is that thequantifiers in HA or PA, ranging over all natural numbers, are not finitary,whereas T is arguably finitary - at least in the sense that T is quantifier free.However, the objects of T are at least prima facie infinitary, and so there is thedifficult question of how to gauge this tradeoff. One idea is that the objects of Tshould not be construed as infinite completed totalities, but rather as rules. Werefer the interested reader to the rather extensive Introductory notes to (Gödel58) in (Gödel 1986-2003 Vol. II).Another application is to extend the interpretation to the two sorted firstorder system known as second order arithmetic, or Z 2 . This was carried out byClifford Spector in (Spector 1962). Here the idea is that one may construe such apowerful extension of Gödel’s Dialectica interpretation as some sort ofconstructive consistency proof for the rather metamathematically strong andhighly impredicative system Z 2 . However, in various communications, Gödelwas not entirely satisfied that the quantifier free system Spector used was trulyconstructive.