Göran Sundholm: Intended interpretations and the choice of formal systems: Themes from Kreisel’s work on constructivist foundations

Abstract: In his lecture at the 1983 Salzburg Gödel Conference, Kreisel noted:

  > The occasion demands a topic that spans, if possible, Gödel’s
working life.
  >
  > For me that singles out the area of intuitionistic logic.

Mutatis mutandis the same applies to the present occasion and Kreisel. Accordingly, my talk deals with Kreisel’s thought on Constructivist. Foundations that provided the setting for much of his most trenchant work, up to and including the period of Hochleistung, 1955–1975. Kreisel’s insightful Bucharest Question from LMPS 4,

  > Was the (logical) language of the current intuitionistic systems obtained by uncritical transfer from languages which were, tacitly, understood classically?

will be used to turn the tables on him with respect to four topics:

(1) Completeness theorems and internal validity [JSL 1958 and 1962];
(2) The Theory of Constructions [Stanford LMPS , 1962 and Saaty 1965];
(3) “Second clauses” [LMPS Stanford and LMPS Amsterdam];
(4) The Theory of the Creating Subject [1967]

All four cases exhibit straightforward carry-over from classical originals, and often mediated by Gödel.

Time permitting, I also consider Kreisel’s much discussed Dictum, as formulated by Michael Dummett, and put on record Kreisel’s own views on this Dictum as given to me in 1978. Furthermore I hope to consider a suggestion of Hidé Ishiguro as to the identity of “Al Tajtelbaum, New York”, the winner of Miss Anscombe’s Analysis Problem No. 10 in Analysis 17, pp. 49–52.