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Home , Consistency, Solomon Feferman

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 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 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 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 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 . which the provable 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 . 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 , the question whether not himself willing to commit to that proposed there exists an 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 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 : (λx.t[x])s t[s/x]. be called a , 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. So he was reduced to chain of equivalences [21]. attending Church’s lectures, which he found pon- In Church’s 1937 review of Turing’s paper, he derous and excessively precise; by contrast, Tur- wrote: ing’s native style was rough-and-ready and prone to minor errors, and it is a question whether As a matter of fact, there is involved Church’s example was of any benefit in this respect. here the equivalence of three different They met from time to time, but apparently there notions: computability by a Turing ma- were no sparks, since Church was retiring by na- chine, general recursiveness in the sense ture and Turing was somewhat of a loner. of Herbrand-Gödel-Kleene, and λ- In the spring of 1937, Turing worked up for pub- definability in the sense of Kleene and lication a proof in greater detail of the equivalence the present reviewer. Of these, the first of machine computability with λ-definability [22]. has the advantage of making the iden- He also published two papers on group , in- tification with effectiveness in the or- cluding one on finite approximations of continu- dinary (not explicitly defined) sense ev- ous groups that was of interest to von Neumann ident immediately… The second and (cf. [24]). Luther P. Eisenhart, who was then head third have the advantage of suitability of the mathematics department, urged Turing to for embodiment in a system of sym- stay on for a second year and apply again for the bolic logic.4 Procter fellowship (worth US$2,000 p.a.). This time, Thus was born what is now called the Church- supported by von Neumann who praised his work Turing Thesis, according to which the effectively on almost periodic functions and continuous computable functions are exactly those computable groups, Turing succeeded in obtaining the fellow- by a Turing machine.5 The (Church-)Turing Thesis ship, and so decided to stay the extra year and do is of course not to be confused with Turing’s the- a Ph.D. under Church. Proposed as a thesis topic sis under Church, our main subject here. was the idea of ordinal logics that had been broached in Church’s course as a way to “escape’’ Turing in Princeton Gödel’s incompleteness theorems. On Newman’s recommendation, Turing decided to Turing, who had just turned 25, returned to spend a year studying with Church, and he ap- England for the summer of 1937, where he de- plied for one of Princeton’s Procter fellowships. In voted himself to three projects: finishing the com- the event, he did not succeed in obtaining it, but putability/λ-definability paper, ordinal logics, and even so he thought he could manage on his fel- the Skewes number. As to the latter, Littlewood had lowship funds from King’s College of 300 pounds shown that π(x) − li(x) changes sign infinitely per annum, and so Turing came to Princeton at the often, with an argument by cases, according to end of September 1936. The Princeton mathemat- whether the Riemann Hypothesis is true or not; ics department had already been a leader on the prior to that it had been conjectured that American scene when it was greatly enriched in the π(n) < li(n) for all n, in view of the massive nu- early 1930s by the establishment of the Institute merical evidence into the billions in support of for Advanced Study. The two shared Fine Hall until that.6 In 1933 Skewes had shown that li(n) <π(n) 1940, so that the lines between them were blurred for some n<103(34) (triple exponential to the and there was significant interaction. Among the base 10) if the Riemann Hypothesis is true. Turing mathematical leading lights that Turing found on hoped to lower Skewes’ bound or eliminate the his arrival were Einstein, von Neumann, and Weyl Riemann Hypothesis; in the end he thought he had at the Institute and Lefschetz in the department; succeeded in doing both and prepared a draft but the visitors that year included Courant and Hardy. did not publish his work.7He was to have a recur- In logic, he had hoped to find—besides Church— ring interest in the R.H. in the following years, in- Gödel, Bernays, Kleene, and Rosser. Gödel had in- cluding devising a method for the practical com- deed commenced a second visit in the fall of 1935 putation of the zeros of the Riemann zeta function but left after a brief period due to illness; he was as explained in the article by Andrew R. Booker in not to return until 1939. Bernays (noted as Hilbert’s this issue of the Notices. Turing also made good collaborator on his consistency program) had progress on his thesis topic and devoted himself

4Church’s review appeared in J. Symbolic Logic 2 (1937), 6li(x) is the (improper) integral from 0 to x of 1/logx and 42–43. is asymptotic to π(x), the number of primes

1202 NOTICES OF THE AMS VOLUME 53, NUMBER 10 full time to it when he returned to Princeton in the The Thesis: Ordinal Logics10 fall, so that he ended up with a draft containing the What Turing calls a logic is nowadays more main results by Christmas of 1937. But then he usually called a formal system, i.e., one prescribed wrote Philip Hall in March 1938 that the work on by an effective specification of a language, set of his thesis was “proving rather intractable, and I am and rules of inference. Where Turing used always rewriting part of it.”8 Later he wrote that “L” for logics I shall use “S” for formal systems. “Church made a number of suggestions which re- Given an effective description of a sequence Snn∈N sulted in the thesis being expanded to an appalling (N ={0, 1, 2,...}) of formal systems all of which length.” One can well appreciate that Church would share the same language and rules of inference, one not knowingly tolerate imprecise formulations or can form a new system Sω = Sn (n ∈ N), by tak- proofs, let alone errors, and the published version ing the effective union of their axiom sets. If the shows that Turing went far to meet such demands sequence of Sn ’s is obtained by iterating an effec- while retaining his distinctive voice and original tive passage from one system to the next, then ways of thinking. Following an oral exam in May, that iteration can be continued to form Sω+l, … and on which his performance was noted as “Excel- so on into the transfinite. This leads to the idea of lent”, the Ph.D. itself was granted in June 1938. Tur- an effective association of formal systems Sα with ing made little use of the doctoral title in the fol- ordinals α. Clearly that can be done only for de- lowing years, since it made no difference for his numerable ordinals, but to deal with limits in an position at Cambridge. But it could have been use- effective way, it turns out that we must work not ful for the start of an academic career in America. with ordinals per se, but with notations for ordinals. Von Neumann thought sufficiently highly of his In 1936, Church and Kleene [3] had introduced a mathematical talents to offer Turing a position as system O of constructive ordinal notations, given his assistant at the Institute. Curiously, at that by certain expressions in the λ-calculus. A variant time von Neumann showed no knowledge or ap- of this uses numerical codes a for such expressions ∈ preciation of his work in logic. It was not until and associates with each α O a countable ordi- | | 1939 that he was to recognize the fundamental im- nal α . For baroque , 1 was taken as the a portance of Turing’s work on computability. Then, notation for 0, 2 as a notation for the successor of |a|, and 3 • 5e for the limit of the sequence |a |, toward the end of World War II, when von Neumann n when this sequence is strictly increasing and when was engaged in the practical design and develop- e is a code of a computable function ˆe with ment of general purpose electronic digital com- ˆe(n)=a for each n ∈ N. The least ordinal not of puters in collaboration with the ENIAC team, he was n the form |a| for some a ∈ O is the analogue, in to incorporate the key idea of Turing’s universal terms of effective computability, of the least un- computing machine in a direct way.9 countable ordinal ω1 and is usually denoted by Von Neumann’s offer was quite attractive, but CK ω1 , where “CK” refers to Church and Kleene. By Turing’s stay in Princeton had not been a person- ∗ an ordinal logic S = Saa∈O is meant any means ally happy one, and he decided to return home de- of effectively associating with each a ∈ O a formal spite the uncertain prospects aside from his fel- system Sa . Note, for example, that there are many lowship at King’s and in face of the brewing rumors ways of forming a sequence of notations an whose of war. After publishing the thesis work he did no limit is ω, given by all the different effectively more on that topic and went on to other things. Not computable strictly increasing subsequences of N. long after his return to England, he joined a course CK So at limit ordinals α<ω1 we will have infinitely at the Government Code and Cypher School, and many representations of α and thus also for its suc- that was to lead to his top secret work during the cessors. An ordinal logic is said to be invariant if war at Bletchley Park on breaking the German whenever |a|=|b| then Sa and Sb prove the same Enigma Code. This fascinating part of the story is theorems. told in full in Hodges’ biography [16], as is his sub- In general, given any effective means of passing sequent career working to build actual comput- from a system S to an extension S of S, one can ∗ ers, promote , theorize about form an ordinal logic S = Saa∈O which is such morphogenesis, and continue his work in mathe- that for each a ∈ O and b = 2a the successor of =  matics. Tragically, this ended with his death in a, Sb Sa , and is further such that whenever e 1954, a probable suicide. a = 3 • 5 then Sa is the union of the sequence of Sˆe(n) for each n ∈ N. In particular, for systems 8Hodges [16], p. 144. whose language contains that of Peano Arithmetic 9  Its suggested implementation is in the Draft report on the PA, one can take S to be S ∪{ConS }, where ConS EDVAC put out by the ENIAC team and signed by von Neu- mann; cf. Hodges [16], pp. 302–303; cf. also ibid., p. 145, 10The background to the material of this section in Gödel’s for von Neumann’s appreciation by 1939 of the significance incompleteness theorems is explained in my piece for the of Turing’s work. Notices [11].

NOVEMBER 2006 NOTICES OF THE AMS 1203 formalizes the consistency statement for S; the for the time; actually, as shown by Georg Kreisel associated ordinal logic S∗ thus iterates adjunction years later, it can even be expressed in ∀-form.12 of consistency through all the constructive ordinal On the other hand, Turing’s class of number- notations. If one starts with PA as the initial sys- theoretical problems does not include such state- tem it may be seen that each Sa is consistent and ments as finiteness of the number of solutions of  so Sa is strictly stronger than Sa by Gödel’s second a diophantine equation (∃∀) or the statement of incompleteness theorem. The consistency state- Waring’s problem (∀∃∀). ments are expressible in ∀ (“for all”)-form, i.e., In section 4 Turing introduced a new idea that ∀xR(x) where R is an effectively decidable predi- was to change the face of the general theory of com- ∗ cate. So a natural question to raise is whether S putation (also known as recursion theory) but the is complete for statements of that form, i.e., only use he made of it there was curiously inessen- whether whenever ∀xR(x) is true in N then it is tial. His aim was to produce an arithmetical prob- provable in Sa for some a ∈ O. Turing’s main re- lem that is not number-theoretical in his sense, i.e., sult for this ordinal logic was that that is indeed not in ∀∃-form. This is trivial by a diagonalization the case, in fact one can always choose such an a argument, since there are only countably many ef- with |a| = ω +1. His ingenious method of proof fective relations R(x, y) of which we could say that was, given R, to construct a sequence ˆe(n) that de- ∀x∃yR(x, y) holds. Turing’s way of dealing with notes n as long as (∀x ≤ n)R(x) holds and that this, instead, is through the new notion of com- • e jumps to the successor of 3 5 when putation relative to an oracle. As he puts it: (∃x ≤ n)¬R(x) .11 Let b =3• 5e and a =2b . Now if ∀xR(x) is true, b ∈ O with |b| = ω. In Sa we can Let us suppose that we are supplied as follows: if ∀xR(x) were not true then Sb with some unspecified means of solv- would be the union of systems that are eventually ing number-theoretical [i.e., ∀∃] prob- the same as Sa, so Sb would prove its own consis- lems; a kind of oracle as it were. … With tency and hence, by Gödel’s theorem, would be in- the help of the oracle we could form a consistent. But Sa proves the consistency of Sb, so new kind of machine (call them o- we must conclude that ∀xR(x) holds after all. machines), having as one of its funda- Turing recognized that this completeness proof mental processes that of solving a given is disappointing because it shifts the question of number-theoretic problem. whether a ∀-statement is true to the question He then showed that the problem of determining whether a number a actually belongs to O. In fact, whether an o-machine terminates on any given the general question, given a, is a ∈ O?, turns out input is an arithmetical problem not computable to be of higher logical complexity than any arith- by any o-machine, and hence not solvable by the metical statement, i.e., one formed by the unlim- oracle itself. Turing did nothing further with the ited iteration of universal and existential quanti- idea of o-machines, either in this paper or after- fiers, ∀ and ∃. Another main result of Turing’s ∗ ward. In 1944 Emil Post [20] took it as his basic no- thesis is that for quite general ordinal logics, S can’t be both complete for statements in ∀-form tion for a theory of degrees of unsolvability, cred- and invariant. It is for these reasons that above I iting Turing with the result that for any set of called his completeness results equivocal. Even so, natural numbers there is another of higher degree what Turing really hoped to obtain was complete- of unsolvability. This transformed the notion of ness for statements in ∀∃ (“for all, there exists”)- computability from an absolute notion into a rel- form. His reason for concentrating on these, which ative one that would lead to entirely new develop- he called “number-theoretical problems”, rather ments and eventually to vastly generalized forms than considering arithmetical statements in general, of recursion theory. Some of the basic ideas and is not clear. This class certainly includes many results of the theory of effective reducibility of number-theoretical statements (in the usual sense the membership problem for one set of numbers of the word) of mathematical interest, e.g., those, to another inaugurated by Turing and Post are ex- such as the twin prime conjecture, that say that an plained in the article by Martin Davis in this issue effectively decidable set C of natural numbers is of the Notices. infinite. Also, as Turing pointed out, the question There are further interesting suggestions and whether a given program for one of his machines asides in the thesis, such as consideration of possi- computes a total function is in ∀∃-form. Of spe- ble constructive analogues of the Continuum Hy- cial note is his proof ([23], sec. 3) that the Riemann pothesis. Finally (as also mentioned by Barry Cooper Hypothesis is a number-theoretical problem in Tur- in his review article), it contained provocative spec- ing’s sense. This was certainly a novel observation ulations concerning intuition versus technical

11Note that e is defined in terms of itself; this is made pos- 12A relatively perspicuous representation in that form sible by Kleene’s index form of the recursion theorem. may be found in Davis et al. [6] p. 335.

1204 NOTICES OF THE AMS VOLUME 53, NUMBER 10 ingenuity in mathematical reasoning. The relevance, P  through O for which there are true sentences according to Turing is that: in ∀-form not provable along that path, as shown in my paper with Spector [12]. The recent book In- When we have an ordinal logic, we are exhaustibility [13] by Torkel Franzén contains an in a position to prove number-theoretic theorems by the intuitive steps of rec- accessible introduction to [7], and his article [14] ognizing [natural numbers as notations gives an interesting explanation (shorn of the off- for ordinals] …We want it to show quite putting details) of what makes Turing’s and my clearly when a step makes use of intu- completeness results work. ition and when it is purely formal… It The problem raised by Turing of recognizing must be beyond all reasonable doubt which expressions (or numbers) are actually nota- that the logic leads to correct results tions for ordinals is dealt with in part through the whenever the intuitive steps [i.e., recog- concept of autonomous progressions of , ob- nition of ordinals] are correct. tained by imposing a boot-strap procedure. That allows one to go to a system Sa only if one already This Turing had clearly accomplished with his for- has a proof in a previously accepted system Sb mulation of the notion of ordinal logic and the ∈ ∗ that a O (or that a recursive ordering of order construction of the particular S obtained by it- type corresponding to a is a well-ordering). Such erating consistency statements. progressions are not complete but have been used One reason that the reception of Turing’s paper to propose characterizations of certain informal may have been so limited is that (no doubt at concepts of proof, such as that of finitist proof Church’s behest) it was formulated in terms of the (Kreisel [18], [19]) and predicative proof (Feferman λ-calculus, which makes expressions for ordinals [8], [9]). and formal systems very hard to understand. He could instead have followed Kleene, who wrote in References his retrospective history [17]: “I myself, perhaps un- [1] A. CHURCH, An unsolvable problem of elementary num- duly influenced by rather chilly receptions from au- ber theory, Amer. J. of Math. 58 (1936), 345–63. diences around 1933–35 to disquisitions on λ- Reprinted in Davis [5]. definability, chose, after general recursiveness had [2] ——— , A note on the Entscheidungsproblem, J. Sym- appeared, to put my work in that format. I cannot bolic Logic 1 (1936), 40–41; correction, ibid., 101–2. complain about my audiences after 1935.” Reprinted in Davis [5]. [3] A. CHURCH and S. C. KLEENE, Formal definitions in the Ordinal Logics Redux theory of ordinal numbers, Fundamenta Mathemati- The problems left open in Turing’s thesis were cae 28 (1936), 11–21. attacked in my 1962 paper, “Transfinite recursive [4] A. M. COHEN and M. J. E. MAYHEW, On the difference progressions of axiomatic theories” [7]. The title π(x) − li(x), Proc. London Math. Soc. 18(3) (1968), contains my rechristening of “ordinal logics” in 691–713; reprinted in Turing [24]. order to give a more precise sense of the subject [5] M. DAVIS, The Undecidable. Basic Papers on Undecid- able Propositions, Unsolvable Problems and Computable matter. In the interests of perspicuity and in order Functions, Raven Press, Hewlett, NY, (1965). to explain what Turing had accomplished, I also re- [6] M. DAVIS, YU. MATIJASEVI˘C, and J. ROBINSON, Hilbert’s cast all the notions in terms of general recursive tenth problem. Diophantine equations: positive as- functions and recursive notions for ordinals rather pects of a negative solution, Mathematical Develop- than the λ-calculus. Next I showed that Turing’s pro- ments Arising From Hilbert Problems, (F. Browder, ed.), gression based on iteration of consistency state- Amer. Math. Soc., Providence, RI, (1976), 323–78. ments is not complete for true ∀∃ statements, [7] S. FEFERMAN, Transfinite recursive progressions of ax- contrary to his hope. In fact, the same holds for the iomatic theories, J. Symbolic Logic 27 (1962), 259–316. even stronger progression obtained by iterating [8] ——— , Systems of predicative analysis, J. Symbolic adjunction to a system S of the local reflection Logic 29 (1964), 1–30. principle for S. This is a scheme that formalizes, [9] ——— , Autonomous transfinite progressions and the for each arithmetical sentence A, that if A is prov- extent of predicative mathematics, Logic, Methodology able in S then A (is true). Then I showed that a pro- and III, (B. van Rootselaar and gression S(U) based on the iteration of the uniform J. F. Staal, eds.), North-Holland, Amsterdam (1968), 121–35. reflection principle is complete for all true arith- [10] , Turing in the land of O(z) , in Herken [15], metical sentences. The latter principle is a scheme ——— 113–47. that formalizes, given S and a formula A(x) that if [11] , The impact of the incompleteness theorems ∀ ——— A(n) is provable in S for each n, then xA(x) (is on mathematics, Notices Amer. Math. Soc. 53 (April true). One can also find a path P through O along 2006), 434–9. which every true arithmetical sentence is provable [12] S. FEFERMAN and C. SPECTOR, Incompleteness along in the progression S(U). On the other hand, invari- paths in recursive progressions of theories, J. Symbolic ance fails badly in the sense that there are paths Logic 27 (1962), 383–90.

NOVEMBER 2006 NOTICES OF THE AMS 1205 [13] T. FRANZÉN, Inexhaustibility. A non-exhaustive treat- ment, Lecture Notes in Logic 28 (2004), Assoc. for Symbolic Logic, A. K. Peters, Ltd., Wellesley (distribs.). [14] ——— , Transfinite progressions: A second look at com- pleteness, Bull. Symbolic Logic 10 (2004), 367–89. [15] R. HERKEN (ed.), The Universal Turing Machine. A Half-Century Survey, , Oxford (1988). [16] A. HODGES, Alan Turing: The Enigma, Simon and Schuster, New York, 1983. New edition, Vintage, Lon- don, 1992. [17] S. C. KLEENE, Origins of recursive function theory, Ann. History of Computing 3 (1981), 52–67. [18] G. KREISEL, Ordinal logics and the characterization of informal concepts of proof, Proc. International Con- gress of Mathematicians at Edinburgh (1958), 289–99. [19] ——— , Principles of proof and ordinals implicit in given concepts, Intuitionism and , (J. My- hill et al., eds.), North-Holland, Amsterdam, (1970) 489–516. [20] E. POST, Recursively enumerable sets and their deci- sion problems, Bull. Amer. Math. Soc. 50 (1944), 284–316. [21] A. M. TURING, On computable numbers, with an ap- plication to the Entscheidungsproblem, Proc. London Math. Soc. 42(2) (1936–37), 230–65; correction, ibid. 43 (1937), 544–6. Reprinted in Davis [5] and Turing [25]. [22] ——— , Computability and λ-definability, J. Symbolic Logic 2 (1937), 153–63. Reprinted in Davis [5] and Turing [25]. [23] ——— , Systems of logic based on ordinals, Proc. Lon- don Math. Soc. (2) (1939), 161–228. Reprinted in Davis [5] and Turing [25]. [24] ——— , Pure Mathematics (J. L. Britton, ed.), Collected Works of A. M. Turing, Elsevier Science Publishers, Amsterdam, (1992). [25] ——— , Mathematical Logic (R. O. Gandy and C. E. M. Yates, eds.), Collected Works of A. M. Turing, Elsevier Science Publishers, Amsterdam, (2001).

1206 NOTICES OF THE AMS VOLUME 53, NUMBER 10