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ARBOR Ciencia, Pensamiento y Cultura Vol. 189-764, noviembre-diciembre 2013, a079 | ISSN-L: 0210-1963 doi: http://dx.doi.org/10.3989/arbor.2013.764n6002

EL LEGADO DE ALAN TURING / THE LEGACY OF ALAN TURING

ON TURING’S LEGACY IN EL LEGADO DE TURING EN MATHEMATICAL AND THE LA LÓGICA MATEMÁTICA Y FOUNDATIONS OF MATHEMATICS* LOS FUNDAMENTOS DE LAS MATEMÁTICAS

Joan Bagaria ICREA (InstitucióCatalana De Recerca I Estudis Avançats) and Departament de Lògica, Història i Filosofia de laCiència, Universitat de Barcelona. [email protected]

Citation/Cómo citar este artículo: Bagaria, J. (2013). “On Copyright: © 2013 CSIC. This is an open-access article distributed Turing’s legacy in and the foundations under the terms of the Creative Commons Attribution-Non of mathematics”. Arbor, 189 (764): a079. doi: http://dx.doi. Commercial (by-nc) Spain 3.0 License. org/10.3989/arbor.2013.764n6002

Received: 10 July 2013. Accepted: 15 September 2013.

ABSTRACT: While Alan Turing is best known for his work on RESUMEN: Alan Turing es conocido sobre todo por sus computer science and cryptography, his impact on the general contribuciones a las ciencias de la computación y a la theory of computable functions ( theory) and the criptografía, pero el impacto de su trabajo en la teoría general foundations of mathematics is of equal importance. In this de las funciones computables (teoría de la recursión) y en los article we give a brief introduction to some of the ideas and fundamentos de la matemática es de igual importancia. En este problems arising from Turing’s work in these areas, such as the artículo damos una breve introducción a algunas de las ideas y analysis of the structure of Turing degrees and the development problemas matemáticos surgidos de la obra de Turing en estas of ordinal . áreas, como el análisis de la estructura de los grados de Turing y el desarrollo de las lógicas ordinales.

Keywords: Alan Turing; Foundations of Mathematics. Palabras clave: Alan Turing; Fundamentos de la matemática.

*This is the written version of the talk with the same title delivered at the “International Symposium: The Legacy of Alan Turing”, held at the Fundación Ramón Areces, Madrid, on 23-24 October 2012. 1. PREAMBLE: THE FOUNDATIONS OF MATHEMATICS ... every mathematical problem can be solved. We IN THE 1930'S are all convinced of that. After all, one of the things that attract us most when we apply ourselves to (1862-1943), one of the most a079 a mathematical problem is precisely that within us prominent mathematicians of his time, developed we always hear the call: here is the problem, search On Turing’s legacy in mathematical logic and the foundations of mathematics in the 1920's an ambitious programme for laying the for the solution; you can find it by pure thoughts, foundations of mathematics1. New firm foundations for in mathematics there is no ignorabimus. were much needed in the wake of the discovery of (Hilbert, 1925) several paradoxes involving some of the most basic mathematical notions, such as those of infinite sets, Thus, it appears that Hilbert was convinced that his or definability. At the beginning of his famous address proposed system for the foundation of mathematics, on The Foundations of Mathematics, delivered at the or perhaps some extension of it, could be Hamburg Mathematical Seminar in July 1927, he said (Hilbert, 1928): Complete: Given a formula φ, either T φ or T φ. ⟝ I pursue a significant goal, for I should like to eliminate ⟝ ¬ once and for all the questions regarding the foundations Further, as an essential requirement for the le- of mathematics [...] by turning every mathematical gitimacy of any strong enough for the into a formula that can be concretely foundation of mathematics (hence involving ideal ele- exhibited and strictly derived. (Hilbert, 1927) ments), Hilbert wanted a proof of consistency, a proof that should be obtained by purely finitary means (i.e., The need for formalization and strict derivation of not involving ideal elements), for he believed that mathematical statements according to explicitly stat- only finitary statements are firmly grounded. Thus, ed logical rules was the only way, according to Hilbert, the system should be provably to be able to reason about ideal elements, such as ac- tual infinite sets, while avoiding the paradoxes. To at- Consistent: For no formula φ we have tain this goal, Hilbert proposed a foundation for math- T φ and T φ. ematics that he calledproof theory, whose goal was to And the proof should⟝ be finitistic⟝ ¬ (hence arithmetical). build a formal system consisting of As Hilbert (1925) acknowledged, • A , in which every mathemati- ... my cannot specify a general method cal statement can be expressed by a formula. for solving every mathematical problem; that does not exist. (Hilbert, 1925) • An effectively given list T of formulas, called axi- oms. These include logical , axioms of equality, He, however, does not give any of why and mathematical axioms (includingaxioms of number, this is so. Yet he seemed to believe that, in addition i.e., arithmetical axioms, as well as explicit definitions, to being complete and consistent, some formal sys- and finite and transfinite recursion axioms). tem based on first-order logic and strong enough to encompass all ordinary mathematics could be • A finite ofrules of for deriving new formulas φ from T, writtenT φ, and read “T proves Decidable: There is a definite method, or φ”. In Hilbert's system [Hilbert, 1928] the only rule of ⟝ mechanical process, by which, given any formula inference is the ). φ, it can be determined whether or not φ is The formal system is not arbitrary, for it is intend- provable in the system. ed to model mathematics in the following sense (Hilbert, 1928): The existence of such an effective method (for any given formal system) is known as the The axioms and provable , that is the formulas (the Decision Problem). The that result from this procedure, are images (Abbildung) Problem had already been posed for different formal of the thoughts constituting customary mathematics as it systems by Schröder (1895) and Löwenheim (1915). has developed until now. (Hilbert, 1928) The formulation above was stated in the context of first-order logic by Hilbert and Ackermann in 1928. In a famous passage from his 1925 address On the Let us observe that completeness implies a positive Infinite, Hilbert expressed the conviction of the solv- solution to the Entscheidungsproblem (assuming the ability of all mathematical problems2 (Hilbert, 1925): axioms are given effectively).

2 ARBOR Vol. 189-764, noviembre-diciembre 2013, a079. ISSN-L: 0210-1963 doi: http://dx.doi.org/10.3989/arbor.2013.764n6002 1.1. Incompleteness las in the relevant formal systems are finite sequences of symbols in a countable alphabet, by coding them by In the autumn of 1930, in his retirement address natural numbers, we may assume that f : → {0, 1}. to the Society of German Scientists and Physicians, in The question is thus to define precisely the notion of a079 Königsberg, and in response to the Latin maxim:Igno - ℕ

computable on the natural numbers. Joan Bagaria ramus et ignorabimus (We do not know, and will not know), Hilbert famously asserted: 1.2. Princeton Wir müssen wissen. Wir werden wissen. (We must In the 1930’s, Princeton University was the centre know. We will know.) of mathematical logic in the United States. The In- stitute for Advanced Studies (IAS), created in 1930 Thus, Hilbert was still holding onto the belief that and housed in the same building as the Department every mathematical problem could be solved, pre- of Mathematics, the old Fine Hall, attracted first-rate sumably by deriving it from a complete formal system. mathematicians such as Oswald Veblen and John von Ironically, just the day before, during the Conference Neumann, as well as Albert Einstein. Kurt Gödel vis- on Epistemology held jointly with the Society meet- ited the IAS on several occasions, giving a series of lec- ings, Kurt Gödel had informally announced his incom- tures in 1934 on his incompleteness results, and be- pleteness result, which represented a fatal blow to coming a permanent member in 1940. In Princeton, Hilbert’s program, at least in the form outlined above. Alonzo Church (1903-1995) was the leading figure Gödel’s First Incompleteness asserts that in Logic. Together with his bright students John Bar- kley Rosser and Stephen Kleene, Church developed the λ-calculus, a formal system designed to formalize Theorem 1. Every consistent formal system that con- the intuitive notion of effectively calculable function. tains a small amount of arithmetic is incomplete, i.e., Church and Kleene introduced a of effectively there are formulas φ such that neither T φ nor calculable functions, called λ-definable, and Church T φ. ⟝ formulated the so-called ⟝ ¬ The amount of arithmetic needed is indeed a very Church’s Thesis (First unpublished version, 1934): A small fragment of Peano’s Arithmetic. For instance, function on the natural numbers is effectively calcula- Robinson’s finitely-axiomatizable theoryQ suffices. ble if and only if it is λ-definable. The second incompleteness theorem is even more In the meantime, Gödel had introduced his class of dramatic. computable functions, known as the Gödel-Herbrand Theorem 2. No consistent formal system that contains general recursive functions, and he had presented a moderate amount of arithmetic can prove its own them during his 1934 visit to Princeton. Shortly after- consistency. I.e., the arithmetical statement wards, Kleene showed that the class of λ-definable functions coincides with the class of Herbrand-Gödel CON(T) recursive functions, and also with the class of Kleene’s that expresses the consistency of the system is not recursive functions, now known simply as recursive. provable in the system itself. Formally, Thus, the first published version of Church’s Thesis reads:

Church’s Thesis (1936): A function on the natural num- In this case, the finitely-axiomatizable fragment of bers is effectively calculable if and only if it is recursive. Peano’s Arithmetic known as Σ -Induction suffices. 1 There is a notion of normal form for formulas of the Thus, no reasonable formal system for mathematics λ-calculus, and in 1935, Church had proved the following: (not even for a basic fragment of arithmetic) can be both consistent and complete. Theorem 3 (Church, 1936a). There is no recursive function on the formulas of the λ-calculus such that But how about the Entscheidungsproblem? To an- on any formula C the value is 2 or 1 according as C has swer this, one needs to make precise the notion ofme - a normal form or not. chanical process, which leads naturally to the notion of computable functionon the natural numbers: Given As a consequence, Church obtains the following: a formal system, one needs a f Corollary 4 (Church, 1936b). The answer to the such that for any given formula φ, f(φ) = 1 if φ is prova- Entscheidungsproblem is negative, even for first-order logic. ble in the system, and f(φ) = 0, otherwise. Since formu-

3 ARBOR Vol. 189-764, noviembre-diciembre 2013, a079. ISSN-L: 0210-1963 doi: http://dx.doi.org/10.3989/arbor.2013.764n6002 That is, there is no effectively computable function sequence is infinite, the machine is called circle-free. on formulas C of first-order logic3 whose value is 1 or 0 Turing calls an infinite sequence of natural numbers according as C is valid or not (equivalently, by Gödel’s computable if it can be computed by a circle-free TM. a079 Completeness Theorem, provable or not in the first- • A proof of the existence of a Universal Turing

On Turing’s legacy in mathematical logic and the foundations of mathematics order logic calculus). Machine. But is this really a solution to the Entscheiduns- A Universal Turing Machine (UTM) is a TM that can problem? Church’s solution relies on Church’s Thesis, be used to compute any computable sequence. That which identifies computable and recursive (as well as is, if the UTM U is set in motion with the code of a TM λ-definable) functions. Gödel was not convinced. The M written on the tape, thenU will produce the same problem is why should every computable function be sequence as M. Turing’s UTM has been regarded as recursive? the first description of the stored-program computer. 2. ALAN TURING ENTERS THE SCENE • A proof of the non-computability of the Halting Problem. Alan Matthison Turing (1912-1954)4, aged 22, was elected Fellow at King’s College in Cambridge on the The Halting Problem is the problem of determining basis of a dissertation on the Gaussian Error Func- whether an arbitrary TM is circle-free or not. Turing tion, in which he proved the Central Limit Theorem proves that there is no circle-free TM that will com- (which he didn’t know had already been proved in pute the sequence of all codes of circle-free TM’s. 1922 by J. W. Lindeberg). In the Spring of 1935 Turing Should such a TM exist, there would be a TM M com-

took a course by Max Newman on the Foundations of puting the sequence 1-fn(n), where fn is the sequence Mathematics, including Gödel’s incompleteness theo- computed by the n-th circle-free TM. If m is the code

rems and the Entscheidungsproblem. He immediately of M, then fm(m) = 1 − fm(m), which is impossible. started working on the problem, producing in April of Turing also proves that there is no TM that, given 1936 a manuscript entitled On Computable Numbers, the code of a TM M, will determine whether the se- with an Application to the Entscheidungsproblem.The quence computed by M contains a 0 or not. And the paper consists of: same for 1. In fact, he shows that the computability of • The analysis of an (idealized) human computer, the Halting Problem is equivalent to the existence of in the process of computing a computable function on such TM’s. the natural numbers. • A “proof” that the computable functions (i.e., • The description of a machine analogue of the those computable by a Turing machine) are indeed human computer: the automatic machines, or a-ma- those that “would naturally be regarded as computable”. chines, now known as Turing Machines. Of course, a real (i.e., mathematical) proof of this As is well-known, a Turing Machine (TM) is a device assertion is not possible, and so the given consisting of atape (unbounded at both ends) divided by Turing are of three kinds: into squares, a head that scans (reads), writes, and (a) A direct appeal to intuition. erases a (0 or 1) on a given square of the tape, and a finite list of configurations. The TM performs (b) A proof of the equivalence between two basic operations: (A) a possible change of symbol computable (i.e., computable by a TM) and in the square being scanned, together with a possible effectively calculable (or recursive). change of configuration, and (B) a possible change of (c) Giving examples of a large class of functions scanned square, together with a possible change of which are computable. configuration. The operation being performed at any given time is determined by the current configuration • A negative solution to theEntscheidungsproblem. and the observed symbol in the scanned square. A TM For each TM M, he writes a first-order formula can thus be described completely by a finitetable, de- Un(M) which says: the symbol 0 appears in the se- tailing the operations, and therefore the table, hence quence computed by M. Then he proves that (1) if the TM, can be coded by a natural number. 0 appears on the sequence computed by M, then When set in motion starting from a blank tape, a Un(M) is provable, and that (2) if Un(M) is provable, TM produces a sequence of binary numbers. If the then 0 appears in the sequence computed by M. In this way a negative solution to the Entscheidung-

4 ARBOR Vol. 189-764, noviembre-diciembre 2013, a079. ISSN-L: 0210-1963 doi: http://dx.doi.org/10.3989/arbor.2013.764n6002 sproblem for first-order logic follows from the non- ceding. A logic Lω may then be constructed in which computability of the Halting Problem. the provable are the totality of theorems provable with the help of the logics L, L , L , ... [...] It is hard to overestimate the importance of Turing’s 1 2 Proceeding in this way we can associate a system of a079 paper, both for the foundations of mathematics and logic with any constructive ordinal. It may be asked the theory of computation. With regards to founda- Joan Bagaria whether a sequence of logics of this kind is complete tions, it is worth noticing Gödel’s enthusiastic reaction in the sense that to any problem A there corresponds (see Gödel, 1964). an ordinal α such that A is solvable by means of the logic L . (Turing, 1939) Due to A. M. Turing’s work, a precise and unquestion- α ably adequate definition of the general concept of The following is an example of the formal systems formal system can now be given ... Turing’s work gives (ordinal logics) studied in his thesis. Let T0 be the set of an analysis of the concept of ‘mechanical procedure’ axioms of Peano’s Arithmetic (PA). Given Tk, let Tk+1:= (alias ‘algorithm’ or ‘computation procedure’ or ‘finite Tk + CON(Tk). Then let Tω := Uk<ω Tk. One can continue combinatorial procedure’). ... A formal system can into the transfinite by letting Tω+1 := Tω + CON(Tω), and simply be defined to be any mechanical procedure for so on. By Gödel’s Second Incompleteness Theorem, producing formulas, called provable formulas. Tα+1 is strictly stronger than Tα , whenever Tα is effec- (Gödel, 1964) tively given. But for Tα to be effectively given, the -or dinal α must be computable. This leads to the notion Thus, while Turing (and Chuch, independently) had of ordinal notations, i.e., a computable way to assign given a negative solution to the Hilbert-Ackermann countable ordinals to natural numbers. Let us denote version of the Entscheidungsproblem, hence yet a fur- by |a| the ordinal assigned to the natural number a. ther blow to Hilbert’s Program, he had given for the first time, according to Gödel, a precise definition of The Church-Kleene system O of ordinal notations: the concept of formal system, thus opening the door The number 1 is a notation for the ordinal 0, 2a is a to a possible revision of the Program. notation for the successor of |a| and 3 . 5e is a nota-

tion for the supremum of the sequence |an|, when- 3. TURING’S DOCTORAL DISSERTATION ever the sequence is strictly increasing and e is the

On the recommendation of Max Newman, Turing code of a computable function ê such that ê(n) = an, got accepted as a graduate student at Princeton. In a for each n. letter to Church, Newman writes: Observe that the set O of ordinal notations, with I should mention that Turing’s work is entirely inde- the order given by a

a Є O with |a| = ω + 1 such that Ta φ. The well-known theorem of Gödel [...] shows that The proof of the Theorem is quite⟝ ingenious (see Fe- every system of logic is in a certain sense incomplete, ferman, 2012), but as Turing himself recognized, it is but at the same time it indicates means whereby from 0 disappointing because it is uninformative. Given a1 ∏ a system L of logic a more complete system L’ may be sentence φ, he does find an a such that, if φ is true, obtained. By repeating the process we get a sequence then a Є O with |a| = ω + 1 and T φ. But this just L, L = L’, L = L’ , ... each more complete than the pre- a 1 2 1 shifts the question about the truth⟝ of φ to the ques-

5 ARBOR Vol. 189-764, noviembre-diciembre 2013, a079. ISSN-L: 0210-1963 doi: http://dx.doi.org/10.3989/arbor.2013.764n6002 tion about the number a belonging to O, a question A Turing Degree is an T-equivalence class. Let D that is actually more complex than any arithmetical denote the set of Turing ≡Degrees with the order in- statement. duced by ≤T. a079 The analysis of the structure of D has produced On Turing’s legacy in mathematical logic and the foundations of mathematics 3.1. Oracles many fascinating new ideas and techniques, such as In section 3 of the Thesis, Turing goes on to consider the priority method, and is still one of the central ar- 0 eas of modern . Much is known what he calls number-theoretic statements (i.e., ∏2 ), namely, those of the form x yR(x, y), where R is a about D. For example: each degree is countable; computable relation. However, he does not succeed there are 2 0-many degrees; each degree has only ∀ ∃ ℵ 0 countably-many predecessors; D is not dense; for in proving a similar completeness theorem for ∏2 statements; not surprisingly, for S. Feferman showed every degree a there is a degree b incomparable with in 1962 that this is impossible (Feferman, 1962). Then a; there are 2 0-many incomparable degrees; every ℵ in section 4, and in order to produce concrete exam- two degrees have a least upper bound, but they need 0 not have a greatest lower bound; etc. ples of problems that are more complex than ∏2 , Tur- ing introduces the notion of computation relative to The structure of D is indeed very complex. For one an oracle. thing, every countable partial ordering can be embed- ded in D. Moreover, the first-order theory of D and Let us suppose that we are supplied with some unspecified the set of theorems of second-order arithmetic are means of solving number-theoretic [i.e., ∏ 0] problems; a 2 recursively isomorphic (S. Simpson, 1977). Perhaps kind of oracle as it were. [...] With the help of the oracle we the most outstanding still open problem about D is could form a new kind of machine (call them o-machines), whether it is rigid, that is, whether D has a non-trivial having as one of its fundamental processes that of solving automorphism. a given number-theoretic problem. (Turing, 1939)

Turing then defines precisely the notion of o-ma- 3.3. On intuition and ingenuity chine (for a fixed oracle o) and formulates the Halting In section 11 of the Thesis, entitled “The purpose of Problem for o-machines, showing that it is not com- ordinal logics”, Turing singles out two faculties of math- putable by any o-machine, with an argument entirely ematical reasoning he callsintuition and ingenuity. analogous to the non-computability of the Halting Problem. Since every ∏ 0 statement is easily seen to 2 The activity of the intuition consists in making spon- be equivalent to one of the form “M is a circle-free 0 taneous judgements which are not the result of con- TM”, if o is an oracle that decides every ∏2 formula, then it follows that the Halting Problem for o-ma- scious trains of reasoning. (Turing, 1939) chines is not ∏ 0. 2 Whereas Although the notion of o-machine is not pursued The exercise of ingenuity in mathematics consists in any further in Turing’s Thesis, with hindsight it is per- aiding the intuition through suitable arrangements haps the most important notion introduced in that of propositions, and perhaps geometrical figures or work, for it is the starting point of the study of relative computability. drawings. It is intended that when these are really well arranged of the intuitive steps which are required cannot seriously be doubted. (Turing, 1939) 3.2. Relative computability By the introduction of formal logic, the need for in- The theory of computability relative to an oracle, tuition is was developed by Emil Post (1897-1984) and Kleene into the theory of degrees of unsolvability, also known ... greatly reduced by setting down formal rules for as Turing Degrees. carrying out which are always intuitively valid. (Turing, 1939) A set of natural numbers A is Turing reducible to a set of natural numbers B if A is computable with ora- But since it is not possible to find a formal logic that would eliminate the use of intuition completely, cle B. Written A ≤T B. because of Gödel’s incompleteness, one is naturally Two sets of natural numbers A and B are Turing forced to consider “non-constructive” systems of equivalent, written A B, if and only if A ≤ B and T T logic, such as ordinal logics. With an ordinal logic we B ≤T A. ≡

6 ARBOR Vol. 189-764, noviembre-diciembre 2013, a079. ISSN-L: 0210-1963 doi: http://dx.doi.org/10.3989/arbor.2013.764n6002 are in a position to prove theorems by the intuitive ACKNOWLEDGMENTS steps of recognizing a number as a notation for an The work was partially supported by the Span- ordinal, and the mechanical (yet requiring ingenuity) ish Ministry of Science and Innovation under grant steps of applying the logical rules. a079 MTM2011-25229, and by the Generalitat de Catalun- Further study of ordinal logics, rechristened as ya (Catalan Government) under grant 2009 SGR 187. Joan Bagaria transfinite recursive progressions of transfinite theo- ries was carried out in the 1960’s by Feferman (1962).

NOTES

1 In the 1920’s, many other people collaborated either jointly 3 A language having only two unary predicate symbols, or just one with Hilbert or independently in the development of what is binary relation symbol suffices. now known as Hilbert’s Program, such as P. Bernays, W. Acker- mann, J. Herbrand, or J. von Neumann. 4 We refer to A. Hodges’ excellent biography (Hodges, 1983) of A. Turing for all details about his life. 2 Address delivered in München on 4 June 1925 at a meeting organized by the Westphalian Mathematical Society to honor 5 A unary relation, or predicate, i.e., a set of natural numbers, is Weierstrass. computable if and only if its characteristic function is computable.

REFERENCES Church, A. (1936a). “An unsolvable problem Gödel, K. (1964). “Postscriptum”. In Davis, From Frege to Gödel. A Source Book of elementary number theory”. Amer. J. M. (ed.), The Undecidable. New York: in Mathematical Logic, 1879-1931. of Math, 58 (2), pp. 345-363. Raven, 1965. Harvard University Press, 1967. Church, A. (1936b). “A Note on the Hilbert, D. (1925). “On the Infinite”, English Hilbert, D. and Ackermann, W. (1928). Entscheidungsproblem”. The Journal of translation in J. van Heijenoort (Ed.), Grundzügen der theoretischen Logik. Symbolic logic, 1 (1), pp. 40-41. From Frege to Gödel. A Source Book in Springer-Verlag. Mathematical Logic, 1879-1931. Har- Feferman, S. (1962). “Transfinite recursive vard University Press, 1967. Hodges, A. (1983). Alan Turing, the Enigma. progressions of transfinite theories”. J. New York: Simon and Schuster Inc. Symbolic Logic, 27, pp. 259-316. Hilbert, D. (1928) “Die Grundlagen der Mathematik”. Abhandlungen aus Turing, A. (1939). “Systems of Logic based Feferman, S. (2012). “Turing’s Thesis”. In den mathematischen Seminar der on ordinals”. Proc. London Math. Soc., Alan Turing’s Systems of Logic. The Prin- Hamburgischen Universität 6. English (2), pp. 161-228. ceton Thesis. Edited by Andrew W. Ap- translation in J. van Heijenoort (ed.), pel. Princeton University Press.

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