Systems of Logic Based on Ordinals (1938)

On 23 September 1936 Turing left England on a vessel bound for New York. His destination was Princeton University, where the Mathematics Department and the Institute for Advanced Study combined to make Princeton a leading centre for mathematics. Turing had applied unsuccessfully for a Visiting Fellowship to Princeton in the spring of 1935. When a year later he learned of Church’s work at Princeton on the Entscheidungsproblem, which paralleled his own (see ‘Computable Numbers: A Guide’), Turing ‘decided quite definitely’ to go there. He planned to stay for a year. In mid-1937 the offer of a Visiting Fellowship for the next academic year persuaded him to prolong his visit, and he embarked on a Ph.D. thesis. Already advanced in his academic career, Turing was an unusual graduate student (in the autumn of 1937, he himself was appointed by Cambridge University to examine a Ph.D. thesis). By October 1937 Turing was looking forward to his thesis being ‘done by about Christmas’. It took just a little longer: ‘Systems of Logic Based on Ordinals’ was accepted on 7 May 1938 and the degree was awarded a few weeks later. The following year the thesis was published in the Proceedings of the London Mathematical Society. ‘Systems of Logic Based on Ordinals’ was written under Church’s supervision. His relationship to Turing—whose formalization of the concept of an effective procedure and work on the Entscheidungsproblem was ‘possibly more convincing’ than Church’s own—was hardly the usual one of doctoral supervisor to graduate student. In an interview given in 1984, Church remarked that Turing ‘had the reputation of being a loner’ and said: ‘I forgot about him when I was speaking about my own graduate students—truth is, he was not really mine.’ Nevertheless Turing and Church had ‘a lot of contact’ and Church ‘discussed his dissertation with him rather carefully’. Church’s influence was not all for the good, however. In May 1938 Turing wrote: My Ph.D. thesis has been delayed a good deal more than I had expected. Church made a number of suggestions which resulted in the thesis being expanded to an appalling length. I hope the length of it won’t make it difficult to get it published.