Turing’s Thesis Solomon Feferman n the sole extended break from his life and var- ing in this way we can associate a sys- ied career in England, Alan Turing spent the tem of logic with any constructive or- years 1936–1938 doing graduate work at dinal. It may be asked whether such a Princeton University under the direction of sequence of logics of this kind is com- IAlonzo Church, the doyen of American logi- plete in the sense that to any problem cians. Those two years sufficed for him to complete A there corresponds an ordinal α such a thesis and obtain the Ph.D. The results of the the- that A is solvable by means of the logic sis were published in 1939 under the title “Systems Lα. of logic based on ordinals” [23]. That was the first Using an ingenious argument in pursuit of this systematic attempt to deal with the natural idea of aim, Turing obtained a striking yet equivocal par- overcoming the Gödelian incompleteness of formal tial completeness result that clearly called for fur- systems by iterating the adjunction of statements— ther investigation. But he did not continue that such as the consistency of the system—that “ought himself, and it would be some twenty years before to” have been accepted but were not derivable; in the line of research he inaugurated would be re- fact these kinds of iterations can be extended into newed by others. The paper itself received little at- the transfinite. As Turing put it beautifully in his tention in the interim, though it contained a num- introduction to [23]: ber of original and stimulating ideas and though Turing’s name had by then been well established The well-known theorem of Gödel through his earlier work on the concept of effec- (1931) shows that every system of logic tive computability. is in a certain sense incomplete, but at Here, in brief, is the story of what led Turing to the same time it indicates means Church, what was in his thesis, and what came whereby from a system L of logic a after, both for him and for the subject.1 more complete system L may be ob- tained. By repeating the process we get From Cambridge to Princeton a sequence L, L1 = L ,L2 = L1 … each As an undergraduate at King’s College, Cambridge, more complete than the preceding. A from 1931 to 1934, Turing was attracted to many logic Lω may then be constructed in parts of mathematics, including mathematical logic. which the provable theorems are the totality of theorems provable with the 1I have written about this at somewhat greater length in help of the logics L, L1,L2 , … Proceed- [10]; that material has also been incorporated as an in- troductory note to Turing’s 1939 paper in the volume, Solomon Feferman is emeritus professor of mathematics Mathematical Logic [25] of his collected works. In its bio- and philosophy at Stanford University. His email address graphical part I drew to a considerable extent on Andrew is [email protected]. Hodges’ superb biography, Alan Turing: The Enigma [16]. 1200 NOTICES OF THE AMS VOLUME 53, NUMBER 10 In 1935 Turing was elected a fellow of King’s Col- inference; since the functions to be defined can lege on the basis of a dissertation in probability the- occur on both sides of the equations, this consti- ory, On the Gaussian error function, which con- tutes a general form of recursion. Gödel explained tained his independent rediscovery of the central this in lectures on the incompleteness results dur- limit theorem. Earlier in that year he began to focus ing his visit to the Princeton Institute for Advanced on problems in logic through his attendance in a Study in 1934, lectures that were attended by course on that subject by the topologist M. H. A. Church and his students Stephen C. Kleene and (Max) Newman. One of the problems from New- J. Barkley Rosser. But Gödel regarded general re- man’s course that captured Turing’s attention was cursiveness only as a “heuristic principle” and was the Entscheidungsproblem, the question whether not himself willing to commit to that proposed there exists an effective method to decide, given analysis. Meanwhile Church had been exploring a any well-formed formula of the pure first-order different answer to the same question in terms of predicate calculus, whether or not it is valid in all his λ-calculus—a fragment of a quite general for- possible interpretations (equivalently, whether or malism for the foundation of mathematics, whose not its negation is satisfiable in some interpreta- fundamental notion is that of arbitrary functions tion). This had been solved in the affirmative for rather than arbitrary sets. The “λ” comes from certain special classes of formulas, but the general Church’s formalism according to which if t[x] is an problem was still open when Turing began grap- expression with one or more occurrences of a vari- pling with it. He became convinced that the answer able x, then λx.t[x] is supposed to denote a func- must be negative, but that in order to demonstrate tion f whose value f (s) for each s is the result, 3 the impossibility of a decision procedure, he would t[s/x], of substituting s for x throughout t. In the have to give an exact mathematical explanation of λ-calculus, function application of one expression what it means to be computable by a strictly me- t to another s as argument is written in the form chanical process. He arrived at such an analysis by ts. Combining these, we have the basic evaluation = mid-April 1936 via the idea of what has come to axiom: (λx.t[x])s t[s/x]. be called a Turing machine, namely an idealized Using a representation of the natural numbers computational device following a finite table of in- in the λ-calculus, a function f is said to be structions (in essence, a program) in discrete ef- λ-definable if there is an expression t such that for fective steps without limitation on time or space each pair of numerals n and m, tn evaluates out = that might be needed for a computation. Further- to m if and only if f (n) m. In conversations with more, he showed that even with such unlimited ca- Gödel, Church proposed λ-definability as the pacities, the answer to the general Entscheidungs- precise explanation of effective computability problem must be negative. Turing quickly prepared (“Church’s Thesis”), but in Gödel’s view that was a draft of his work entitled “On computable num- “thoroughly unsatisfactory”. It was only through a chain of equivalences that ended up with Turing’s bers, with an application to the Entscheidungs- analysis that Gödel later came to accept it, albeit problem”; Newman was at first skeptical of Turing’s indirectly. The first link in the chain was forged with analysis but then became convinced and encour- the proof by Church and Kleene that λ-definabil- aged its publication. ity is equivalent to general recursiveness. Thus Neither Newman nor Turing were aware at that when Church finally announced his “Thesis” in point that there were already two other proposals published form in 1936 [1], it was in terms of the under serious consideration for analyzing the gen- latter. In that paper, Church applied his thesis to eral concept of effective computability: one by demonstrate the effective unsolvability of various Gödel called general recursiveness, building on an mathematical and logical problems, including the idea of Herbrand, and the other by Church, in terms decision problem for sufficiently strong formal of what he called the λ-calculus.2 In answer to the systems. And then in his follow-up paper [2] sub- question, “Which functions of natural numbers are mitted April 15, 1936—just around the time Tur- effectively computable?”, the Herbrand-Gödel ap- ing was showing Newman his draft—Church proved proach was formulated in terms of finite systems the unsolvability of the Entscheidungsproblem for of equations from which the values of the functions logic. When news of this work reached Cambridge are to be deduced using some elementary rules of a month later, the initial reaction was great disap- pointment at being scooped, but it was agreed that 2The development of ideas about computability in this period by Herbrand, Gödel, Church, Turing, and Post has Turing’s analysis was sufficiently different to still been much written about and can only be touched on warrant publication. After submitting it for publi- here. For more detail I recommend the article by Kleene cation toward the end of May 1936, Turing tacked [17] and the articles by Hodges, Kleene, Gandy, and Davis in Part I of Herken’s collection [15], among others. One 3One must avoid the “collision” of free and bound vari- of the many good online sources with further links is ables in the process, i.e., no free variable z of s must end at http://plato.stanford.edu/entries/church- up within the scope of a “λz”; this can be done by renaming turing/, by B. J. Copeland. bound variables as necessary. NOVEMBER 2006 NOTICES OF THE AMS 1201 on an appendix in August of that year in which he visited 1935–36, but did not visit the States again sketched the proof of equivalence of computabil- until after the war. Kleene and Rosser had received ity by a machine in his sense with that of λ- their Ph.D.’s by the time Turing arrived and had left definability, thus forging the second link in the to take positions elsewhere.