Abstract
The problem of effectiveness tended early, as we have seen, to be conflated with the paradoxical one of ‘finite definablility’. That the two were really distinct problems emerges from an observation due essentially to Hilbert, that α is finitely definable ⇔ α is definable (period!).
The conclusion that man is not a machine is invalid. E. Post
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Notes
Of course, certain important consequences of Dedekind’s recursion theorem, notably his isomorphism of any two Peano models, have then to be relativized to the universe of the formal set theory. In particular, the result fails when countable models of the theory are considered.
This is discussed in more detail below.
See Borel [29], p. 213.
Borel and du Bois-Reymond often used the diagonal method for the construction of fast-growing functions in growth-rate arguments. See du Bois-Reymond [72] and Borel’s remarks in [166], pp. 12ff. This use of diagonalization is prominent in recursive function theory, especiallv the branch dealing with complexitv and speed of computation.
That is, g(x) = f x (x) must also be a (partial) unary function of (E). The Richard’s Paradox is avoided by the partiality of g.
Perhaps no principle of logical inquiry has been more used yet less discussed in the history of logic from Aristotle to Frege. Thus, as late as Boole, substitution principles are used apparently without any conscious awareness. Even Frege, though he is conscious of using such principles in his axiomatic formulations of propositional and predicate logic, never explicitly lists them as actual rules of inference. Indeed, he claims, for instance, that modus ponens is the only rule of inference in formulation of propositional logic. See [255], p. 3.
See Schönfinkel [255], p. 357; Curry was a student of Hilbert (see below in text).
The application of a combinator x to an argument y is written (xy). When x represents a function, (xy) can be interpreted as the result of substituting y for the first variable in x. Thus, if f represents a unary function, (fx) means what is ordinarily written f(x). (fx) may then be either applied to other arguments y yielding ((fx)y), or taken as an argument of another function g yielding (g(fx)). Application is not associative, so that (g(fx)) is not the same as ((gf)x). The expression fx1 ... x n is often used to abbreviate ((... (fx1) ...)x n ). Abinary function F(x, y) is construed as a unary function f which, when applied to an argument x, has as its value the function (fx) which, when applied to the argument y has F(x, y) as its value, i.e. F(x, y) = ((fx)y), or just fxy for short. Schönfinkel introduced the combinators S and K governed by the rules: Sxyz = xz(yz), Kxy = x. Thus, if x represents a binary function f and y a unary function g, then Sxyz means what is ordinarily written as f(z, g(z)). K is the ‘constant function’: if x represents a constant, K is the constant function whose value is x for every argument y. See Stenlund [240], whose formulation I have largely followed.
See Stenlund [240], pp. 14ff.
See note 8 for an explanation of S and K.
Stenlund remarks that “the difference between the λ-calculus and the theory of combinators is parallel to the situation in set theory where the comprehension axiom can be replaced by class operations” ([240], p. 16).
His original definition was W = SS(SK).
See Curry [58], pp. 184–5 and pp. 154–6 for the interesting history of these developments. To briefly explain this, we need to define the ‘composition’ combinator B = S(KS)K, which will satisfy the rule Bfgx = f(gx). Curry’s original ‘paradoxical combinatory’ Y is then defined by Y = WS(BWB), where W is the diagonalizing combinator defined in the previous note. Y is now called a‘fixed-point’ combinator since it can be proved to satisfy the theorem YX = X(YX), for every X. Thus, given any comnbinator X, Y produces a fixed-point for X. This is the combinatory analogue for the fixed-point theorems of Gödel and Kleene discussed below in the text. In fact, it is the essential construction used by Kleene [147] in his very important proof discussed below that all recursive functions are λ-definable. Bernays [20] has emphasized the importance of these results for the formalist program.
See Ackermann [2], pp. 10ff., and von Neumann [195], pp. 298–9. The essential problem was in establishing sufficient ‘Kontrollierbarkeit’ over the possible recursions admitted as axioms, by means of assigning increasing ordinals to increasingly complex recursive definitions.
This approach led to a certain arbitrariness in the specification of their formal systems for arithmetic. Thus, after introducing all the ordinary Peano-Dedekind axioms, Ackermann continued: Zu diesen ein für allemal festehenden Axiomen können noch Definitions-axiome für Funktionen hinzukommen. Sämtliche Funktionen werden durch Rekursion definiert. Bei der Aufstellung dieser Axiome ist eine gewisse Willkür gestattet. Wir schreiben nur ein Schema vor, in das sich alle diese Axiome einordnen lasse müssen ... Wie viele derartige Axiome man gebraucht ... und welche Gestalt diese in einzelnen haben, ist der Willkür überlassen ([2], p. 5).
This has been claimed, though without explicit textual support, by Steen [237a] and K. Mainzer.
See Péter [203] for an extensive discussion of this theme.
This would have been a fair description of Skolem’s work of 1923 but Hilbert apparently was not acquainted with Skolem’s paper at that time.
If one is really looking for the dream of Hilbert’s which was undermined by Gödel’s work, then this is it. There is nothing in this world he wanted more than to use his prooftheory to solve what he considered the most beautiful of all mathematical problems, Cantor’s continuum problem, but his approach required that there be uncountably many effective functions defined by various recursions. As I shall argue below, when the significance of Gödel’s incompleteness construction began to emerge, especially in the work of Kleene, it pointed inevitably to (CT), viz. to the countability of all effective functions.
In particular, Hilbert’s ‘second lemma’ (see [255], p. 391). This lemma itself, incidentally, is very much in the spirit of (CT), for the latter can be thought of as a‘reducibility’ principle asserting that however complex and high-type function and set quantifiers you use in defining an effective function, this definition can always be reduced to one expressible in a very simple first-order formalism. This theme is discussed below in a more cybernetical context.
See Kalmar [1401.
I shall follow the formulations of Gödel’s 1934 lectures rather than his 1931 paper.
This was proVed simultaneously and independently by Gödel and Gentzen in 1932. See Davis [61], pp. 75–82.
See Hilbert-Bernays [106], pp. 420ff., for a full discussion.
Unless, of course, F is inconsistent.
See Siefkes [233], pp. 8ff.
See Siefkes [233], pp. 45ff.
The following example of such a recursion is given by Rogers [221], who credits it to Kreisel. Where T(x, y, z) is the primitive recursive Kleene-predicate for recursive functions we define: g(x, y, z) = 1’ (p)[T(x, x, p) ∧ p y), 2z, otherwise. If we now add, to a set of recursion equations for g, the additional equation f(x, y) = g(x, y, f(x, y + 1)), it follows, by Kleene’s result, that the predicate (3y)T(x, x, y) is recursively undecidable, that the function fjust defined cannot be recursive. On the other hand, it can be shown that f is the unique solution for the set of equations just described. It follows that the values of f cannot be computed from these equations by any fixed set of effective rules — assuming (CT).
See Gödel [100], p. 71.
See Kleene in [61], p. 259 for details and discussion.
See Kleene [146], Theorem XIII; also Kleene [150], pp. 257, 259.
This shows the fallacy of Kreisel’s claim in [157] that a complete formalization would have established Church’s thesis: a complete formal system for arithmetic in which all recursive functions were definable would lead to an easv refutation of Church’s thesis by the diagonal method, for in such a case the diagonal function would surely be effectively computable, but not even definable in F, if consistent — hence a counterexample to Church’s thesis. The resolution of the Richard paradox effected by Church’s thesis may be seen as ‘dialectical’ in Hegel’s sense. We begin historically with the thesis (RP) (ii) of the countability of the effective functions, but later the diagonal method led to the antithesis of their uncountability. Finally, Church’s thesis has led to a‘synthesis’ of the two: for while it decides on (RP) (ii) as the literal truth, it finds a‘kernel of truth’ in (RP) (i) also, namely that the effective functions are not effectively countable. Curiously, Zermelo [278] regarded Gödel’s non-r.e. set of unprovable sentences as uncountable (cf. [107]). See Webb [261], p. 164 for Church’s temporary identification of these two notions.
T(m, x, y) will then mean that y is the Gödel-number of a computation of the machine with Gödel-number m which began with the input x.
Van Dalen [86], p. 261. This opinion is widespread among intuitionists. Brouwer occasionally spoke of ‘finite algorithms’ for calculation but completely ignored (CT). Of course, van Dalen is right: Church’s thesis does seem improbable, as I have tried to bring out in several ways. In fact, Bowie [31] has tried to refute it by arguing that, since any physical device for computing functions by means of a random element might compute any (!?) function at all, the probability that it computes a recursive function is zero. But this argument no more leads to an explicit counterexample than does the intuitionistic metamathematical approach. Indeed, one may regard Bowie’s remark not so much as an argument against Church’s thesis as simply another way of emphasizing what a strong thesis it is. As an argument against the thesis it would be every bit as ironic as the intuitionistic ones, depending as it does on the vague notion of ‘randomness’ whose best explication in some contexts depends precisely on this thesis itself, Lucas [176a], pp. 122–3 claims that only humans and not machines can introduce any element of randomness, but this seems rather implausible in the face of quantum mechanics and undecidability. (cf. Chapter I.) See also Ross [223] for a good critical discussion of Bowie’s argument, and Berg and Chihara [15] for some misunderstandings which both Bowie and Ross fall prey to.
Kreisel [156], p. 128.
I am following van Dalen [86]. Troelstra [250] contains an excellent discussion of some serious problems besetting the interpretation of I⊢- n A. The axioms (CS) are generally not themselves added to the intuitionistic number-theory formalisms ; rather their principal role is that of justifying the Kripke schema given below, which follows from (CS) and the following function definition by intuitionistic logic. The axiom (CS) (i) is alsosupposed to insure that this definition is effective - assuming, of course, the intuitionistic meaning of disjunction. But this seems to beg the question. In any event, this definition does not lead to any effective non-recursive functions after all.
See Troelstra [250], pp. 100ff.
See Troelstra [250], pp. 100ff.
See Church’s remarks in Davis [61], p. 100.
See Strong [241] for an extensive survey of efforts in this direction. See also Bird [25] for a good introduction to the related development, stemming largely from D. Scott, of basing the whole theory of computation on Kleene’s fixed-point and recursion theorems. Thomas [248] defuses Kleene’s argument for (CT) based on them.
See Takeuti [243], where the recursion theorem is employed to sharpen the consistency proofs for number theory and sub-systems of analysis. The key step in such proofs is to replace the ordinary formal induction axioms with the effective ω-rule and then show that cuts can always be effectively removed from the resulting infinite proof-trees. (Con sistency then follows trivially from this cut-elimination property.) Takeuti makes a rather complicated use of the recursion theorems to prove the existence of a recursive function which, applied to any (description of an) infinite proof-tree, always eventually yields a (description of a) cut-free infinite proof. Transfinite induction then has to be used to prove that the Turing machine computing this function always halts, i.e., that this function is total.
See Minsky [184], pp. 106ff. for an illuminating discussion of the problem this posed for early investigators of effectiveness.
See Turner [253], p. 86 for a good critical discussion of this assumption.
[181], p. 35.
Post did not say where this criticism came from. Kreisel [157] claimed that Gödel would soon publish a “forthcoming note” in Dialectica on his objections to Turing’s arguments for (T)′, but it has not appeared. Meanwhile Wang [259] quotes and paraphrases Gödel extensively on the matter. In particular, it contains a long note, from which we quote in the following, that Wang says will soon appear appended to Gödel’s 1934 lectures in a new edition of [61].
See Wang [259], p. 326.
It is not clear to me why Gödel’s mental states would not also become confused in the eventuality of (G)′, for even if mind exists separately from matter, our thought is seemingly ‘channeled’ through our brains. This is the classical cartesian problem of the interaction between mind and body.
See Turing [61], p. 117; also Minsky [184], pp. 15ff.
See Wang [259], p. 326.
As examples, we take M1 ={qo1Bq0} , M2 = q01 . For the quadruples of our multiplier M 7 , see Davis [60], pp. 12ff. We assume that each machine begins in the starting state q o , scanning the leftmost stroke 1 on its tape. An input of n is represented by n + 1 strokes in successive cells of the tape, which we call an “n-block”. An input of n and m to, say, our adder M 3 , is represented by an n-block and an mn-block on the tape separated by a single blank ‘B’. An output of n is just n (possibly scattered) l’s on the tape. Naturally, each of these machines performs very different tasks if they begin in arbitrary states on arbitrary parts of arbitrary tapes. Also they may contain finite auxilliary alphabets.
Thus, e.g., we can imagine, as would be plausible, that M6 was designed to double numbers, i.e. to compute ∅6(x) = 2x. However, it will compute something no matter how many separated input-blocks we put on its tape: with 2 such inputs, M 6 computes ∅ 6 (x, y) = 2x +y + 1. But, in general, we would not initially be interested in these ‘by-products’ of M6.
Thus, e.g., if we fix a 5-block on M3’s tape, it can be used to compute ∅3(5, x) = 5+ x. Note, however, that we do not have ∅3(x) = 5+ x but rather ∅3(x) = x, as the reader m ay easily verify by inspecting the quadruples given for M 3 . And if we fix a 2block on M7’s tape we can use it to compute ∅7(2, x) = 2x =∅ 6 (x), for by definition we have ∅ 7 (x, y) = x• y.
That is, we know we can compute u(z, x) effectively when it is defined. Note that we always have ∅ Z (x) defined ⇔ M[x + 1] halts, where the right side is short for the machine Mz, started in state q o scanning the leftmost stroke on a tape containing only an x + 1 block, eventually halts. Also Z (x, y) defined ⇔ M Z [x + 1B y+ 1] halts.
See Moore [187] and Singh [234], pp. 185–6 for a discussion of this problem.
See Minsky [184], pp. 106ff.
He did both already in 1924!
Ibid., p. 424. This anticipates Kreisel [156].
Chihara [51], p. 562, makes a similar point very neatly.
Whether or not (10) brings out any physicallv significant feature is quite another matter. In fact, Wigner [273] has used quantum mechanical calculations to argue against the physical plausibility of von Neumann’s [196] results — which did not use (10). Specifically, Wigner tries to show that a self-reproducing system depends, from the quantum mechanical point of view, on the satisfiability of a very large number of equations in relatively few unknowns, so that “it would be a miracle” (ibid., p. 205) if they were satisfied. Arbib [4] points out, however, that one of the assumptions made by Wigner in his calculations — roughly speaking, that self-reproducing systems have no special structure — is very implausible for such systems. But even if Wigner could get his result from biologically reasonable assumptions, it would not be clear that the ‘miracle’ he would see in the satisfiability of his equations is any greater than that of the satisfiability of Kleene’s equations (10). Indeed, the solvability of the latter is closely related to the miracle Gödel saw in the failure of diagonalization to lead outside the recursive functions. See Hofstadter [132] for a revealing discussion of the relation between selfreproducing machines and the molecular biology of self-reproducing systems.
These consequences of Kleene’s (10) may make the following explanation of its significance easier to understand: Now suppose that we are asked to write a program B whose outputs are to depend in a prescribed way not only on ... but also on B itself.... We may object that we cannot reasonably be asked to write a program without knowing in advance what this program is supposed to do; and in the present case what the program is supposed to do depends on the — yet unwritten — program itself. A programmer confronted with such a task might well feel like the wise men of Babylon when the were commanded by Nebuchadnezzar not only to interpret his dream but to guess what the dream was. Like them he might object that this was not the kind of problem he could fairly be expected to solve ([11], p. 274). But this is just the sort of problem that (10) enables us to solve.
In a penetrating note on Church’s thesis, Parikh [196a] points out both the importance and difficulty of isolating a suitable set of atomic acts for generalizing or extrapolating this thesis to cover such seemingly effective everyday tasks as cooking from a recipe.
An exception is Hofstadter [ 132], which also contains a full and useful discussion of Turing’s thesis. I regret that this remarkable book did not appear in time to take fuller advantage of its discussions of certain topics treated herein, particularly that of machine vision.
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Webb, J.C. (1980). Effectiveness Mechanized. In: Mechanism, Mentalism and Metamathematics. Synthese Library, vol 137. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7653-6_4
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