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A complete theory of natural, rational, and real numbers

Published online by Cambridge University Press:  12 March 2014

John R. Myhill
John R. Myhill*
Affiliation:
Temple University
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Extract

The purpose of the present paper is to construct a fragment of number theory not subject to Gödel incompletability. Originally the system was designed as a metalanguage for classical mathematics (see section 10); but it now appears to the author worthwhile to present it as a mathematical system in its own right, to serve however rather as an instrument of computation than of proof. Its resources in the latter respect seem very extensive, sufficient apparently for the systematic tabulation of every function used in any but the most recondite physics. The author intends to pursue this topic in a later paper; the present one will simply present the system, along with proofs of consistency and completeness and a few metatheorems which will be used as lemmas for future research.

Completeness is achieved by sacrificing the notions of negation and universal quantification customary in number-theoretic systems; the losses consequent upon this are made good in part by the use of the ancestral as a primitive idea. The general outlines of the system follow closely the pattern of Fitch's “basic logic”; however the latter system uses combinatory operators in place of the variables used in the present paper, and if variables are introduced into Fitch's system by definition their range of values will be found to be much more extensive than that of my variables. The present system K is thus a weaker form of Fitch's system.

It is apparently not known whether or not Fitch's system is complete.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 15 , Issue 3 , September 1950 , pp. 185 - 196
Copyright
Copyright © Association for Symbolic Logic 1950

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References

1 Fitch, F. B., A basic logic, this Journal, vol. 7 (1942), pp. 105–114Google Scholar.

2 F. B. Fitch, loc. cit., p. 109.

3 The writer is indebted to the referee of a previous version of this article for this neat statement of the relation between his system and Fitch's system.

4 It is not possible to say whether or not Fitch's system is (semantically) complete until one knows just what U-expressions are wffs of that system. Prof. Fitch has expressed the belief that it is semantically complete if one regards wffs as comprising every U-expression of the form ‘[a = b]’ and every expression obtainable from wffs by application of the rules of inference.

5 The idea behind this construction of the natural numbers is due to L. Chwistek. See e.g. The limits of science (1948), pp. 93–95Google Scholar.

6 The metalinguistic notation of this paper follows that of Quine's, W. V.Mathematical logic (1941)Google Scholar, except that (1) variables ‘α’, ‘β’, ‘γ’, etc. do the work of Quine's variables ‘ζ’, ‘η’, etc. as well as of his ‘α’, ‘β’, etc.; (2) variables ‘a’, ‘b’, etc. range over the names of numerical sequences; (3) variables ‘p’, ‘q’, etc. do the work of Quine's ‘ϕ’, ‘Ψ’, etc.; (4) corners are omitted; (5) large Greek letters are introduced in section 8 as variables ranging over large italic letters, which are restricted variables in the object language.

7 The terminology is an extension of W. V. Quine's, loc. cit., p. 152.

8 α need not have a free occurrence in p; i.e., ‘vacuous’ quantification and abstraction are permitted. Cf. W. V. Quine, loc. cit., p. 74 and pp. 177–8.

9 This and the next three clauses are taken with slight modifications from W. V. Quine, loc. cit., pp. 142–3.

10 Cf. W. V. Quine, loc. cit., 1182 (p. 137).

11 Cf. W. V. Quine, loc. cit., *100 (p. 88).

12 Cf. W. V. Quine, ibid.

13 Cf. W. V. Quine, loc. cit., *232 (p. 172).

14 Cf. W. V. Quine, loc. cit., *235 (p. 174) and footnote 16 below.

15 Cf. W. V. Quine, loc. cit., †514 (p. 217).

16 Cf. W. V. Quine, loc. cit., †517 (p. 217). It is evident from a comparison of RR1–7 with the theorems and metatheorems of Quine's system cited in footnotes 10–16 that every theorem of K is a theorem of Quine's system. The only one of RR1–7 whose proof in Quine's system is non-trivial is R5; this requires use of the elementhood assumptions ‘0 ϵ V’ and ‘x, y ϵ V ⊃ x; y ϵ V’ in order to remove the restriction ⌜ζ ϵ V⌝ from *235.

17 Corresponding to one of the meanings of L. Chwistek's multi-purpose star-operator (cf. the construction of finite relations in The limits of science, pp. 97–8, and that of rationale in loc. cit., pp. 99–100).

18 Cf. W. V. Quine, loc. cit., pp. 201–2.

19 L. Chwistek, loc. cit., p. 94; and cf. p. 133.

20 Fitch, F. B., A basic logic, p. 107Google Scholar, Rule [=]; and cf. idem, An extension of basic logic, this Journal, vol. 13 (1948), p. 96, Rule [=].

21 In fact, were it not for the seeming strangeness of a rule of inference requiring a varying number of premises, we could have replaced RR6–7 by the following: if ((a 1; a 2) ϵ α), ((a 2; a 3) ϵ α), …, ((a n−1; a n) ϵ α) are theorems, so is ((a 1; a n) ϵ *α). The proof of the equivalence of this rule to RR6–7 is trivial; see Fitch, F. B., An extension of basic logic, p. 97Google Scholar, Rule [*] and the sentence beginning at the bottom of p. 97; and compare idem, A basic logic, p. 108, Rule [*].

22 For ‘=’ as primitive and classically interpreted, see e.g. Hilbert, D. and Bernays, P., Grundlagen der Mathematik, (1934–1939), vol. 1, p. 165–9Google Scholar. For the other symbols, see W. V. Quine, loc. cit., pp. 11, 12–13, 119–20 and 130–32 respectively.

23 Cf. D. Hilbert and P. Bernays, loc. cit., vol. 2, pp. 9–18.

24 The same feature probably characterizes certain set-theoretic systems containing a very strong form of the axiom of choice. For let a relation < well-order the values of the variable ‘x’; then ‘(Ex)(… x—)’ is true if and only if ‘… ŷ(Ez)(… z—.y ϵ z.(w)(… w—⊃ (w = z ∨ z < w)))—’ is true. I am indebted to Dr. Ilse Novak for this observation.

25 Cf. W. V. Quine, loc. cit., pp. 215–16, 220–21.

26 For the exponent notation see W. V. Quine, loc. cit., pp. 253–54.

27 This is obvious if we consider the alternative formulation of RR6–7 given in footnote 21.

28 ‘True’ without qualification will henceforth mean ‘true with reference to II1–8’.

29 Cf. footnote 27.

30 Cf. F. V. Fitch, A basic logic, p. 58.

31 Cf. W. V. Quine, loc. cit., p. 202, D25.

32 Cf. W. V. Quine, loc. cit., pp. 197–8.

33 Cf. W. V. Quine, loc. cit., p. 200, D24.

34 Cf. Fitch, F. B., A basic logic, p. 58Google Scholar.

35 See Section 2, and 12, Section 5.

36 Cf. Fitch, F. B., An extension of basic logic, p. 103Google Scholar, definition of ‘Nat’; and W. V. Quine, loc. cit., p. 241, D40.

37 Cf. Fitch, F. B., A basic logic, p. 107Google Scholar, Abbreviations; and W. V. Quine, loc. cit., p. 48, D6.

38 The variables ‘w’, ‘x’, ‘y’, ‘z’, and their subscripted variants are restricted to natural numbers throughout the informal discussion of this section.

39 (5) is usually given in the more general form: (5′) Any function f such that

f(0, x 2, …, xn) = g(x 2, …, xn),f(x 1 + 1, x 2, …, xn) = h(x 1,f(x 1, x 2, …, xn), where g and h are primitive recursive functions; see e.g. Fitch, F. B., Representations of calculi, this Journal, vol. 9, (1944), p. 58Google Scholar. However, it has been shown by Péter, R. (Über den Zusammenhang der verschiedenen Begriffe der rekursiven Funktion, Mathematische Annalen, vol. 110, pp. 612–632CrossRefGoogle Scholar) that every function definable by the schemata (l)–(4) and (5′) is definable by the schemata (l)–(5).

40 Cf. F. B. Fitch, loc. cit., p. 59.

41 Since α and the β's by hypothesis designate number-theoretic relations, we are under no obligation to use restricted variables; and the present notation facilitates the introduction of D12 in a general form applicable also to non-number-theoretic situations.

42 Kleene, S. C., Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53, p. 56, Theorem VGoogle Scholar.

43 S. C. Kleene, loc. cit., p. 63. Kleene defines a formal deductive theory for a relation R as a theory in which all true formulae of the form R(a 1, …, a n), and no false formula of the form is provable, (loc. cit., p. 61.)

44 says that x = y z; the notation is from Peano (cf. Quine, loc. cit., p. 259), who uses it in a slightly different but related sense.

45 Fitch, F. B., A basic logic, p. 105Google Scholar.

46 Rosser, B., Extensions of some theorems of Gödel and Church, this Journal, vol. 1 (1935), p. 88, Lemma I, Corollary IGoogle Scholar.

47 See D. Hilbert and P. Bernays, loc. cit., vol. 1, p. 371.

48 See Fitch, F. B., Representations of calculi, pp. 57–62Google Scholar.

49 Tarski, A., Der Wahrheitsbegriff in den formalisierten Sprachen, Studia philosophica, vol. 1, pp. 261–405Google Scholar.

50 Fitch, F. B., Representations of calculi, p. 61Google Scholar.

51 For the notation see F. B. Fitch, A basic logic.

52 The class Fmla, as well as the relations Nom and Subst mentioned below, are clearly recursive and therefore, by Section 9, definable in K. Cf. Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198CrossRefGoogle Scholar.

53 S. C. Kleene, loc. cit., p. 63, Theorem VIII.

54 α is vacuous; cf. W. V. Quine, loc. cit., p. 178, *254.

55 The definitions are almost exactly those in W. V. Quine, loc. cit., pp. 235–6. (D18, D19, D27, D28, D29, D44–45 respectively. In D44 ‘(S'γ)’ ‘ must of course give way to ‘ (γ ; γ)’.)

56 Cf. D12.

57 Since writing the above the writer has seen an article by O. Specker, this Journal, vol. 14 (1949), p. 145, which probably contains theorems equivalent to his Theorems I and II, and proofs or disproofs of statements equivalent to his Conjectures I and II.

58 The writer would like to take this opportunity of thanking Professor F. B. Fitch of Yale University for many ideas, both in his published writings and in correspondence.