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Systems of transfinite types involving λ-conversion

Published online by Cambridge University Press:  12 March 2014

Maurice L'Abbé
Maurice L'Abbé*
Affiliation:
Université De Montréal
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Extract

The problem of constructing formal systems involving transfinite types was briefly suggested many times in the literature. Actually few attempts have been made to develop such systems. In particular, there exists a system based on a set of axioms and rules of inference suggested by Church and worked out in detail by E. Bustamante in his dissertation. Recently, John Kemeny has reformulated this system in a simplified way and has obtained an interesting hierarchy of systems. It so happens that all these systems considered by Kemeny may be regarded as successive extensions of a particular formulation of the theory of finite types, namely a certain monadic functional calculus of order ω whose main ideas are due to Tarski.

There exists on the other hand a quite different formulation of the theory of finite types which is due to Church. It is a formulation which incorporates certain features of the calculus of λ-conversion, and which has revealed itself quite advantageous for certain purposes. The formulation of the theory of transfinite types which we are going to introduce in this paper may be considered as an attempt to generalize Church's formulation of the theory of finite types.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 18 , Issue 3 , September 1953 , pp. 209 - 224
Copyright
Copyright © Association for Symbolic Logic 1953

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References

2 See for instance the following references: Hilbert, David, Über das Unendliche, Mathematische Annalen, vol. 95 (1926), p. 183 ffCrossRefGoogle Scholar.; Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), p. 191Google Scholar; Tarski, Alfred, Der Wahreitsbegriff in der formalisierten Sprachen, Studia Philosophica, I (1936), p. 393 ff.Google Scholar; Tarski, Alfred, On undecidable statements in enlarged systems of logic and the concept of truth, this Journal, vol. 4 (1939), p. 110Google Scholar.

3 Bustamante, Enrique, Transfinite type theory, Thesis 1944, PrincetonGoogle Scholar. Another formulation of transfinite type theory was made by Frank Bruner in a monograph entitled Mathematical logic with transfinite types, 1943 (see IX 72). However this last work diverges in content from what we have here in mind.

4 Kemeny, John, Type theory and set theory, Thesis 1949, PrincetonGoogle Scholar. Some of the results included in that thesis were announced at the meeting of the Association for Symbolic Logic in December, 1949 (cf. this Journal, vol. 15 (1950), p. 78).

5 Tarski, Alfred, Einige Betrachtungen über die Begriffe der ω-Widerspruchsfreiheit und der ω-Vollständigkeit, Monatshefte für Mathematik und Physik, vol. 40 (1933), pp. 97–103CrossRefGoogle Scholar.

6 Church, Alonzo, A formulation of the simple theory of types, this Journal, vol. 5 (1940), pp. 56–68Google Scholar.

7 The choice of the basic type symbols is quite arbitrary, except that they must form a set which can be effectively well-ordered, i.e. it must be possible to well-order that set by a certain relation ≺ in such a way that there exists an effective method to decide for any two elements a and b of the set whether we have a ≺ a or b ≺ a.

8 Henkin, Leon, Completeness in the theory of types, this Journal, vol. 15 (1950), p. 83 ffGoogle Scholar.

9 Loc. cit., p. 60.

10 The notation is used in the following to denote the result of substituting Bβ for all occurrences of aβ in Mα.

11 Concerning the reasons for choosing that particular function c, see the remarks following the definition of Φ021 in the next section.

12 To indicate that A 0 is a theorem of Σ0 or of Σ2, we will write ⊢0A 0 or ⊢2A 0, respectively.

13 Following Church (see XIV 197) the notation [A, B, C] is used below in M3, and later on, as an abbreviation for (B⊃A) & (~B ⊃ C).

14 Another way to deal with this problem is to reduce this complex recursion to the ordinary one by using a technique described in Hilbert, and Bernays, , Grundlagen der Mathematik, vol. 1 (1934), p. 326–328Google Scholar.

15 This is sufficient in view of Gödel's famous theorem.

We notice that the result we are going to establish in this section proves, furthermore, that our system does not fall under a certain objection which can be raised against certain possible formulations of the theory of transfinite types. This objection, which can be found in Turing, A., Systems of logic based on ordinals, Proceedings of the London Mathematical Society, s. 2, vol. 45 (1939), pp. 197–207Google Scholar, amounts to showing that, in certain formulations, no theorem of arithmetic for instance can be proved which could not already be proved in the similar formulation containing only finite types. Our system Σ2 however (and similarly for the general systems Σν later introduced) escapes that objection because it includes axioms of an essentially different content than those included in the original system of finite types Σ0, namely the axioms A16.