my forty years on his shoulders - Department of Mathematics

35We believe that the Spector development has not been fully exploited. Inparticular, it ought to give rather striking mathematically interestingcharacterizations of the provably recursive functions and provable ordinals of Z 2and various fragments of Z 2 .Another fairly recent application is to use the Dialectica interpretation,and extensions of it to systems involving functions and real numbers, in order toobtain sharper uniformities in certain areas of functional analysis that had beenobtained before by the specialists. This work has been pioneered by U.Kohlenbach. See the five references to Kohlenbach (and joint authors) in the listof references.7. THE AXIOM OF CHOICE AND THE CONTINUUMHYPOTHESIS.Gödel wrote six manuscripts directly concerned with the continuumhypothesis: Two abstracts, (Gödel 1938), (Gödel 1939a). One paper with sketchesof proofs, (Gödel 1939b). One research monograph with fully detailed proofs,(Gödel 1940). One philosophical paper, (Gödel 1947,1964), in two versions.The normal abbreviations for the axiom of choice is AxC. The normalabbreviation for the continuum hypothesis is CH.A particularly attractive formulation of CH asserts that every set of realnumbers is either in one-one correspondence with a set of natural numbers, or inone-one correspondence with the set of real numbers.Normally, one follows Gödel in considering CH only in the presence ofAxC. However, note that in this form, CH can be naturally considered withoutthe presence of AxC. However, Solovay’s model satisfying ZFCD + “all sets are