Abstract
The conceptual confluence of Post’s and Turing’s analysis of combinatory processes, respectively of mechanical procedures, is the central topic in Davis and Sieg’s [14]. Where Turing argued convincingly for the adequacy of his notion of machine computation in 1936, Post viewed his identical notion in the same year as being tied to a working hypothesis in need of “continual verification”. Post gave novel and informative arguments for his thesis or, as he put it, generalization . He insisted, however, that ultimately a psychological analysis “of mental processes involved in combinatory mathematical processes” has to be given. In this way, he hoped to obtain a natural law and thus the basis for the claim that the undecidability and incompleteness theorems constitute “a fundamental discovery in the limitations of the mathematizing power of Homo Sapiens”. Our detailed analysis of (the background for) his work on the issues leads to an unambiguous answer to the question Did Post have Turing’s Thesis?: He did [not].
...I study Mathematics as a product of the human mind and not as absolute...
(Post in Anticipation, i.e., (Post [47], p. 64)).
This paper was written for Martin Davis, brilliant mathematical logician, expert computer scientist, and dedicated student of Post; his humanity and devotion to logic are transcendent.—For Wilfried Sieg, he has been a mentor and friend for more than thirty years; during the last few years we have been collaborating, e.g., organizing a session for the Turing Centenary Conference in Cambridge in 2012 and writing a joint paper, our [14]. It has been a pleasure and privilege to do so.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Church [2] had obtained, also in 1936, the same unsolvability result as Turing. He used Gödel’s recursive functions, which he knew to be equivalent to the \(\lambda \)-definable ones, as the precise notion of computability.
- 2.
- 3.
C. I. Lewis’s influence on Post is discussed also in (Urquhart [60], pp. 619–620).
- 4.
The completeness theorem for sentential logic had been established in its modern formulation in Bernays’s [2] Habilitationsschrift of 1918, for the same calculus of Principia Mathematica. The independence investigations that are also contained in that work were published in (Bernays [3]).—Post’s [43] book on Iterative Systems originated as a “companion piece” to his dissertation. There he determined “all the non-equivalent sub-languages of the language of the complete two-valued propositional calculus”. (Post [43], p. 3) For a contemporary view of this highly interesting work, see (Urquhart [60], Sect. 5).
- 5.
There is an interesting difference between their motivations. Łukasiewicz mentions in his [35, 36] from 1918 and 1920, respectively, that he was led to the idea of a third truth-value while working on antinomies and “the principle of contradiction in Aristotle’s work” (Łukasiewicz [35], p. 86). In order to avoid antinomies, the “third logical value may be interpreted as possibility” (Łukasiewicz [35], p. 87). Post, in contrast, does not give any interpretation of the multiple truth-values except for the purely mathematical one; he treats the corresponding truth tables as syntactic objects similar to their treatment in the two-valued case.
- 6.
The emergence of categoricity and completeness is described in (Awodey and Reck [1]).
- 7.
The purely mathematical part reducing a combinatorial system (in form C) to a normal one had already been published in 1943 under the title Formal reductions of the general combinatorial decision problem. The long final note of this paper gives a “brief résumé” of the larger developments of which this mathematical investigation was a part and which are fully described in (Post [47]).
- 8.
The motivation for canonical form B , but also the familiar concepts of reduction and equivalence are discussed in detail on pages 7–8 of Anticipation and, similarly, in (Post [45], p. 289).
- 9.
Indeed, full equivalence of *10 and systems in canonical form seems to have been a goal; see footnotes 26, 79, and 90 of Anticipation . Thus, the solvability of the finiteness problem of any of the systems would imply solvability for all the others, in particular for predicate logic.—Footnote 79 indicates that the reduction of systems in normal form to those of form A, the so-called closing of the circle, was not done in 1920–21, but only later.
- 10.
A version of this reduction, much easier to understand than Post’s, is given in (Minsky [37]) and was further refined in (Szabó [55]).—For Post, as pointed out in (De Mol [18], p. 53), this result supported his conjecture that all of Principia Mathematica could be reduced to a normal system ; he wrote in (Anticipation , p. 45): “...for if the meager formal apparatus of our final normal systems can wipe out all the additional greater complexities of canonical form B , the more complicated machinery of the latter should clearly be able to handle formulations correspondingly more complicated than itself.”—To emphasize the significance of this result we quote a pregnant remark from Minsky ([37], p. 240): “We have long felt that this result is one of the most beautiful theorems in mathematics. The fact that any formal system can be reduced to Post canonical systems with a single axiom and productions of the restricted [normal] form is in itself a remarkable discovery, and even more so when we learn that this was found in 1921, long before the formalization of metamathematics became so popular.”.
- 11.
Kleene was convinced by the same argument; see (Kleene [33], p. 59). In (Post [45], p. 285) “overwhelming evidence” is adduced for Church’s Thesis by reference to footnote 2 in (Kleene [31]). There one finds a concise and masterful summary of the evidence, as Kleene saw it, for the “identification” of effective calculability and recursiveness; Kleene’s remarks are quoted in footnote 34, below.
- 12.
The tag systems are special normal systems: the g’s are all of the same length; each \(g'\) depends only on the first letter of its corresponding g.—The important role of the problem of “tags” was emphasized in §3 of Anticipation; see also the very careful analysis in (De Mol’s [16, 17]). Indeed, De Mol argues that the tag systems were crucial for Post for two main reasons: (1) they prompted his belief that there might be absolutely unsolvable problems, and (2) they inspired the formulation of normal systems.
- 13.
The character of this argument and its similarity to “absoluteness” considerations of Gödel, Church, Hilbert & Bernays, and Turing will be discussed briefly in footnote 29 of Sect. 7.7 below.—Martin Davis pointed out that the above considerations of course do not establish the undecidability of predicate logic. Post is very cautious on what he claims to have established with respect to that problem; see footnotes 79 and 90 of Anticipation and also footnote 10 above.
- 14.
The full sentence is this: “Our remaining ‘theorems’ deserve that name only in the sense that a complete mathematical proof thereof clearly can be given—as contrasted with our generalization of §7”.
- 15.
The footnotes were added by Post at the time of writing the Anticipation paper in the late 1930s and early 1940s. They were not part of his original notes from the 1920s.
- 16.
That is, Post remarks in note 96, b serves as a negation symbol.
- 17.
This explains why the requirement on L was not weakened in case (S, P) is an assertion of L: “There is no reason for doing so since by suitably adjoining K to such a weak L the stronger L would result.” (Anticipation, p. 53).
- 18.
Gödel formulated the first incompleteness theorem in its full generality as pertaining to all consistent formal systems containing some elementary number theory most strikingly in 1964 in his [26], the Postscriptum to his Princeton Lecture Notes. For the “precise and unquestionably adequate” characterization of formal systems he appealed to Turing’s and Post’s work. He wrote there: “A formal system can simply be defined by any mechanical procedure for producing formulas, called provable formulas.”.
- 19.
At this very spot Post remarks (Anticipation, p. 55) that his conclusion goes contrary to the viewpoint of C. I. Lewis as it was reported above in Sect. 7.2. Furthermore, he mentions that it is “not so much contrary to Russell’s viewpoint (since he does not fully express himself)” and that it is “in line with Bergson’s Creative Evolution”. Post is not correct with his remark on Lewis, as the latter had a more sophisticated understanding of mathematics than expressed through the programmatic heterodox view; see (Lewis [34], pp. 359–361).
- 20.
One particularly fitting meaning for counterpart is articulated as follows: “One of two parts which fit and complete each other; a person or a thing forming a natural complement to another.”.
- 21.
The various features mentioned in the informal discussion found their way into the definition of a “creative set” of natural numbers in (Post [45], p. 295). Post envisions on page 296 not only a finite, but indeed transfinite iteration of this extending process. The iteration along Kleene’s constructive ordinals is actually carried out in (Davis [9], p. 190) continuing work in (Davis [8]). The statement added to a particular system S is the Gödel sentence G for S. Assuming that S satisfies the standard representability and derivability conditions, G is equivalent to the consistency statement for S. The results, for this type of extension, from Feferman’s [20] progressions of theories (and Turing’s [57] ordinal logics) can be directly transferred to the Post-Davis construction.
- 22.
Clearly, Post’s discussion excludes sets definable by generalized inductive definitions, like Kleene’s O, as generated sets, as they require a “rule” with infinitely many premises.
- 23.
Turing, in his illuminating and informal paper from 1954, entitled Solvable and unsolvable problems, formulates the thesis not for mechanical procedures or generated sets, but rather for puzzles as follows: “The normal form for puzzles is the substitution type of puzzle [i.e., a particular kind of Post canonical system].” He remarks then, “The statement is moreover one which one does not attempt to prove. ... for its status is something between a theorem and a definition. In so far as we know a priori what is a puzzle and what is not, the statement is a theorem. In so far as we do not know what puzzles are, the statement is a definition which tells us something about what they are.” (Turing [59], p. 15) As puzzles can be given “finite coordinates”, they are more general syntactic configurations. It should be mentioned that Post also considered broader classes of syntactic configurations; see (Urquhart [60], p. 643).
- 24.
Kleene, straddling Post’s and Church’s positions, wisely remarked in his [32], the classical Introduction to metamathematics, “While we cannot prove Church’s thesis, since its role is to delimit precisely an hitherto vaguely conceived totality, we require evidence that it cannot conflict with the intuitive notion which it is supposed to complete; i.e. we require evidence that every particular function which our intuitive notion would authenticate as effectively calculable is general recursive. The thesis may be considered a hypothesis about the intuitive notion of effective calculability, or a mathematical definition of effective calculability; in the latter case, the evidence is required to give the theory based on the definition its intended significance.” (Kleene [32], pp. 318–9).
- 25.
In footnote 12 of Anticipation, Post asserts that “the bubble of symbolic logic as universal logical machine finally [has] burst” on account of the undecidability and incompleteness results; he adds, “Actually, the old dream of symbolic logic is finding partial realization in Tarski’s recent work on decision problems .”
- 26.
In the very footnote in which Post articulates this difference, he also asserts that the first goal has been achieved by the work in §7. The contributions to the proposed complete analysis, needed to achieve the second goal, are fragmentary. They are sometimes quite obscure and difficult to grasp, in particular, those related to the “analysis of proof” with the goal of finding an absolutely undecidable proposition. See Anticipation footnotes 4 and 6 as well as the remarks on the “process of proof” starting on page 59. Post writes, the limitations in man’s mathematical powers “suggest that in the realms of proof ... a problem may be posed whose difficulties we can never overcome; that is that we may be able to find a definite proposition which can never be proved or disproved.” (Anticipation, p. 56) Then he refers back to footnote 1 in which he describes, more expansively, a “fundamental problem”, namely, the question of “the existence of absolutely undecidable propositions which in some a-priori fashion can be said to have a determined truth-value, and yet cannot be proved or disproved by any valid logic.” (Anticipation, p. 1) That is, of course, in striking opposition to the rationalist optimism of Hilbert and Gödel that is beautifully expressed in (Gödel [25], p. 164).
- 27.
From Latin ‘digesta’ (n., pl.) meaning ‘Matters methodically arranged’.
- 28.
Recall from Sect. 7.2 the non-semantic understanding of truth and falsity.
- 29.
Post’s argument for this assertion resembles Gödel’s for the absoluteness of the notion of computable functions in his [24, 26]. A similar argument for identifying the notion of calculable functions with recursiveness is found in Church’s letter to Pepis from June 8, 1937, which is reprinted in the Appendix of (Sieg [49]). Each argument shows that, as long as broad informal conditions are satisfied, the extensions of particular kinds don’t allow for more computations than the restricted frameworks. Considerations of the same kind are found in Supplement II of Hilbert & Bernays’s Grundlagen der Mathematik II as well as in (Turing [56], Sect. 9, II).—This notion of “absoluteness”, obviously quite different from Post’s, is discussed in (Sieg [51], 572–7). We should point out that Post’s argument suffers from the same kind of subtle circularity as Gödel’s and Church’s, because it is required that the extensions have to have postulates and rules of the same general form as those of Principia Mathematica.
- 30.
- 31.
Turing is discussed in notes 6, 9, 112, 118, and 120 of Anticipation. Post most strongly emphasizes the role of the “finite number of mental states hypothesis”. However, why would its correctness make Post’s position (as he remarks in note 9) “largely academic”? And, why would it make (as he says in note 6) “the detailed development envisioned in the Appendix unnecessary”? - Post is grappling with a different problem; in addition, it is not just the number of mental states that is important for Post’s considerations, but also their internal elementary, discrete structure. (Wider formulations, as stressed in [41], are to achieve greater psychological fidelity.).
- 32.
Note in particular that the internal configurations of machines or m-configurations, which correspond to the states of mind of the human computer, are directly incorporated into the symbolic configurations on which the (Post) production rules operate.
- 33.
See (Sieg [52]) for the discussion of this variant of Turing’s Thesis introducing a normal forms for puzzles; see also footnote 23.
- 34.
For the reader’s convenience, we quote the essential points of Kleene’s footnote, which is attached to the statement, “A function of natural numbers, with natural numbers as values, is taken to be effective if it is Herbrand-Gödel recursive.” Here is the (partial) quote: “...This notion of effectiveness appears, on the following evidence, to be general. A variety of particular effective functions and classes of effective functions (selected with the intention of exhausting known types) have been found to be recursive. Two other notions, with the same heuristic property, have been proved to be equivalent to the present one, viz., Church-Kleene \(\lambda \)-definability and Turing computability. Turing’s formulation comprises the functions computable by machines. ... Functions determined by algorithms and by the derivation in symbolic logics of equations giving their values (provided the individual steps have an effectiveness property which may be expressed in terms of recursiveness) are recursive.”.
- 35.
We contended in Sect. 7.7, turning “Post’s analysis on its head”, that one can extract from his analysis, when stripped of philosophical preconceptions and reversed, an argument that is strikingly similar to Turing’s.
- 36.
This has, however, significant repercussions on the very mathematical powers: they are limited, but also creative (in the restrictive sense we pointed out at the end of Sect. 7.5). That was made clear by Post, when asserting, “mathematicians are better than machines”, as they can prove theorems machines cannot. Such a proof requires not only an argument outside the system, but for the creative extension of the system it also requires, that the extending statement be recognized as true. See Anticipation, footnotes 12 and 100.
- 37.
Post finished his graduate education in 1920; Keyser’s book was published only in 1922. Yet it is most likely that Post was familiar with Keyser’s views expressed in the book. In the Preface (page vii) Keyser mentions that the book is the result of more than forty years of reflection on the nature of mathematics.
- 38.
The underlying distinction between structural and formal axiomatics is discussed in Sieg [54].
References
Awodey, Steve and Reck, Erich. 2002. “Completeness and Categoricity, Part II: 20th Century Metalogic to 21st Century Semantics.” History and Philosophy of Logic 23, no. 2: 77–94.
Bernays, Paul. 1918. Beiträge zur axiomatischen Behandlung des Logik-Kalküls. Habilitations-schrift. Göttingen. Reprinted in [20], 222–68.
Bernays, Paul. 1926. “Axiomatische Untersuchung des Aussagen-Kalküls der ‘Principia Mathematica’.” Mathematische Zeitschrift 25, 305–320.
Church, Alonzo. 1935. “An Unsolvable Problem of Elementary Number Theory. Preliminary Report (Abstract).” Bulletin of the American Mathematical Society 41: 332–33.
Church, Alonzo. 1936. “An Unsolvable Problem of Elementary Number Theory.” The American Journal of Mathematics 58, no. 2: 345–63.
Church, Alonzo. 1937. “Review of (Post, 1936).” The Journal of Symbolic Logic 2, no. 1: 43.
Church, Alonzo. 1938. “The Constructive Second Number Class.” Bulletin of the American Mathematical Society 44, 224–32.
Davis, Martin. 1950. On the Theory of Recursive Unsolvability. Dissertation, Princeton University.
Davis, Martin. 1958. Computability and Unsolvability. New York: McGraw-Hill.
Davis, Martin. 1982. “Why Gödel Didn’t Have Church’s Thesis.” Information and Control 54, no. 1–2: 3–24.
Davis, Martin. 1994. “Emil L. Post: His Life and Work.” In [14], xi–xxviii.
Davis, Martin, ed. 1965. The Undecidable. Hewlett, New York: Raven Press. Reprinted by Dover Publications, Mineola, N.Y., 2004.
Davis, Martin. 1994. Solvability, Provability, Definability: The Collected Works of Emil L. Post. Basel: Birkhäuser.
Davis, Martin and Wilfried Sieg. 2016. “Conceptual Confluence in 1936: Post & Turing.” In Turing Centenary Volume edited by Thomas Strahm and Giovanni Sommaruga. Basel: Birkhäuser, 3–27.
Dedekind, Richard. 1888. Was sind und was sollen die Zahlen? Braunschweig: Vieweg.
De Mol, Liesbeth. 2006. “Closing the Circle. An Analysis of Emil Post’s Early Work.” The Bulletin of Symbolic Logic 12, no. 2: 267–289.
De Mol, Liesbeth. 2011. “On the complex behaviour of simple tag systems. An experimental Approach.” Theoretical Computer Science 412, no. 1–2: 97–112.
De Mol, Liesbeth. 2013. “Generating, Solving and the Mathematics of Homo Sapiens. Emil Post’s Views on Computation.” In A Computable Universe – Understanding and Exploring Nature as Computation, edited by Hector Zenil. New Jersey: World Scientific, 45–62.
Ewald, William and Wilfried Sieg, eds. 2013. David Hilbert’s Lectures on the Foundations of Arithmetic and Logic, 1917–1933. Heidelberg: Springer.
Feferman, Solomon. 1962. “Transfinite recursive progressions of axiomatic theories.” The Journal of Symbolic Logic 27: 259–316.
Gandy, Robin. 1988. “The confluence of ideas in 1936.” In The Universal Turing Machine – A Half-century Survey, edited by Rolf Herken. Oxford: Oxford University Press, 55–111.
Gödel, Kurt. 1931. “On Formally Undecidable Propositions of Principia Mathematica and Related Systems I.” Translated by Elliott Mendelson. In [13], 5–38.
Gödel, Kurt. 1934. “On Undecidable Propositions of Formal Mathematical Systems.” In [13], 41–71.
Gödel, Kurt. 1936. “On the Length of Proofs.” In [13], 82–3.
Gödel, Kurt. 193?. “[Undecidable Diophantine Propositions].” In Kurt Gödel. Collected Works, edited by Solomon Feferman, et. al. Vol. 3. Oxford: Oxford University Press, 164–75.
Gödel, Kurt. 1964. “Postscriptum to (Gödel, 1934).” In [13], 71–3.
Gödel, Kurt. 1972. “Some remarks on the undecidability results.” In Kurt Gödel. Collected Works, edited by Solomon Feferman, et. al. Vol. 2. Oxford: Oxford University Press, 305–6.
Gödel, Kurt. 2003. “Correspondence with Emil Post.” In Kurt Gödel. Collected Works, edited by Solomon Feferman, et. al. Vol. 5. Oxford: Oxford University Press, 169–74.
Keyser, Cassius. 1912. “Principia Mathematica.” Science ser. II, vol. 35, no. 890: 106–10.
Keyser, Cassius. 1922. Mathematical philosophy, a study of fate and freedom; lectures for educated laymen. New York: E. P. Dutton & Company.
Kleene, Stephen Cole. 1938. “On Notation for Ordinal Numbers.” Journal of Symbolic Logic 3, no. 4: 150–55.
Kleene, Stephen Cole. 1952. Introduction to Metamathematics. Groningen: Elsevier.
Kleene, Stephen Cole. 1981. “Origins of Recursive Function Theory.” Annals of the History of Computing 3, no. 1: 52–67.
Lewis, Clarence Irving. 1918. A Survey of Symbolic Logic. Berkeley: University of California Press.
Łukasiewicz, Jan. 1918. “Farewell Lecture.” In Selected Works of Jan Łukasiewicz, edited by Ludwik Borkowski. Amsterdam: North-Holland Publishing Company. 1970. 84–6.
Łukasiewicz. 1920. “On Three-Valued Logic.” In Selected Works of Jan Łukasiewicz, edited by Ludwik Borkowski. Amsterdam: North-Holland Publishing Company. 1970. 87–8.
Minsky, Marvin. 1967. Computation: Finite and Infinite Machines. London: Prentice-Hall.
Post, Emil. 1921. “Introduction to a General Theory of Elementary Propositions.” American Journal of Mathematics 43, no. 3: 163–85.
Post, Emil. 1921. “On a Simple Class of Deductive Systems. (Abstract)” In Solvability, Provability, Definability: The Collected Works of Emil L. Post. Basel: Birkhäuser. 545.
Post, Emil. 1930. “Generalized differentiation.” Transactions of the American Mathematical Society 32: 723–81.
Post, Emil. 1936. “Finite Combinatory Processes – Formulation 1.” The Journal of Symbolic Logic 1, no. 3: 103–5.
Post, Emil. 1940. “Polyadic groups.” Transactions of the American Mathematical Society 48: 208–350.
Post, Emil. 1941. The Two-Valued Iterative Systems of Mathematical Logic. Princeton. Princeton University Press.
Post, Emil. 1943. “Formal Reductions of the General Combinatorial Decision Problem.” American Journal of Mathematics 65, no. 2: 197–215.
Post, Emil. 1944. “Recursively Enumerable Sets of Positive Integers and Their Decision Problems.” American Mathematical Society 50, no. 5: 284–316.
Post, Emil. 1947. “Recursive unsolvability of a problem of Thue.” Journal of Symbolic Logic 12: 1–11.
Post, Emil. 1965. “Absolutely Unsolvable Problems and Relatively Undecidable Propositions - Account of an Anticipation.” In [12], 375–441.
Sieg, Wilfried. 1994. “Mechanical procedures and mathematical experience.” In Mathematics and Mind, edited by Alexander George. Oxford: Oxford University Press, 71–117.
Sieg, Wilfried. 1997. “Step by Recursive Step: Church’s Analysis of Effective Calculability”. The Bulletin of Symbolic Logic 3, no. 2: 154–180. Reprinted in 2014. (with a Postscriptum) in Turing’s Legacy: Developments from Turing’s ideas in logic, edited by Rod Downey. Cambridge: Cambridge University Press, 434–66.
Sieg, Wilfried. 2002. “Calculations by man and machine: conceptual analysis.” In Reflections on the foundations of mathematics, edited by Wilfried Sieg, Richard Sommer, and Carolyn Talcott. Natick: A. K. Peters, 390–409.
Sieg, Wilfried. 2009. “On Computability.” In Philosophy of Mathematics (Handbook of the Philosophy of Science) edited by Andrew Irvine. Amsterdam: North-Holland Publishing Company, 535–630.
Sieg, Wilfried. 2012. “Normal forms for puzzles: A variant of Turing’s Thesis.” In Alan Turing – His work and Impact, edited by S. Barry Cooper and Jan van Leeuwen. Amsterdam: Elsevier, 332–8.
Sieg, Wilfried. 2013. “Gödel’s Philosophical Challenge (to Turing).” In Computability, edited by Jack Copeland, Carl Posy, and Oron Shagrir. Cambridge: MIT Press, 183–202.
Sieg, Wilfried. 2014. “The ways of Hilbert’s axiomatics: structural and formal.” Perspectives on Science 22, no. 1: 133–57.
Szabó, Máté. 2014. Post and Kalmár on Turing and Church. MS Thesis, Carnegie Mellon University, Department of Philosophy.
Turing, Alan. 1936. “On computable numbers with an application to the Entscheidungsproblem.” Proceedings of the London Mathematical Society, 2, vol. 42: 230–65.
Turing, Alan. 1939. “Systems of logic based on ordinals.” Proceedings of the London Mathematical Society, ser. 2, 45: 161–228.
Turing, Alan. 1950. “The word problem in semi-groups with cancellation.” The Annals of Mathematics, Ser. 2, Vol. 52, no. 2: 491–505.
Turing, Alan. 1954. “Solvable and unsolvable problems.” Science News 31: 7–23.
Urquhart, Alasdair. 2009. “Emil Post.” In Handbook of the History of Logic. Logic from Russell to Church, edited by Dov M. Gabbay and John Woods. Vol. 5. Amsterdam: North–Holland Publishing Company. 429–78.
Whitehead, Alfred and Bertrand Russell. 1910. Principia Mathematica. Vol. 1. Cambridge: Cambridge University Press.
Acknowledgements
We thank Martin Davis, Liesbeth De Mol, and Ulrik Buchholtz for encouraging, but also critical remarks. Liesbeth’s detailed comments prompted real changes in our presentation. We are also very grateful to Alberto Policriti and Eugenio Omodeo , who as editors of this volume were extremely patient with our difficulties completing this essay.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Sieg, W., Szabó, M., McLaughlin, D. (2016). Why Post Did [Not] Have Turing’s Thesis. In: Omodeo, E., Policriti, A. (eds) Martin Davis on Computability, Computational Logic, and Mathematical Foundations. Outstanding Contributions to Logic, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-41842-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-41842-1_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-41841-4
Online ISBN: 978-3-319-41842-1
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)