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Completeness in the theory of types1

Published online by Cambridge University Press:  12 March 2014

Leon Henkin*
Affiliation:
Princeton University

Extract

The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system.

For the functional calculus of second order, in which predicate variables may be bound, a very different kind of result is known: no matter what (recursive) set of axioms are chosen, the system will contain a formula which is valid but not a formal theorem. This follows from results of Gödel concerning systems containing a theory of natural numbers, because a finite categorical set of axioms for the positive integers can be formulated within a second order calculus to which a functional constant has been added.

By a valid formula of the second order calculus is meant one which expresses a true proposition whenever the individual variables are interpreted as ranging over an (arbitrary) domain of elements while the functional variables of degree n range over all sets of ordered n-tuples of individuals. Under this definition of validity, we must conclude from Gödel's results that the calculus is essentially incomplete.

It happens, however, that there is a wider class of models which furnish an interpretation for the symbolism of the calculus consistent with the usual axioms and formal rules of inference. Roughly, these models consist of an arbitrary domain of individuals, as before, but now an arbitrary class of sets of ordered n-tuples of individuals as the range for functional variables of degree n. If we redefine the notion of valid formula to mean one which expresses a true proposition with respect to every one of these models, we can then prove that the usual axiom system for the second order calculus is complete: a formula is valid if and only if it is a formal theorem.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1950

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Footnotes

1

The material in this paper is included in The completeness of formal systems, a Thesis presented to the faculty of Princeton University in candidacy for the degree of Doctor of Philosophy and accepted in October, 1947. The results were announced at the meeting of the Association for Symbolic Logic in December, 1947 (cf. this Journal, vol. 13 (1948), p. 61).

References

3 Gödel, Kurt, Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 349360CrossRefGoogle Scholar.

4 Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173198CrossRefGoogle Scholar.

5 These classes cannot really be taken in an altogether arbitrary manner if every formula is to have an interpretation. For example, if the formula F(x) is interpreted as meaning that x is in the class F, then ˜F(x) means that x is in the complement of F; hence the range for functional variables such as F should be closed under complementation. Similarly, if G refers to a set of ordered pairs in some model, then the set of individuals x satisfying the formula (∃y)G(x, y) is a projection of the set G; hence, we require that the various domains be closed under projection. In short, each method of compounding formulas of the calculus has associated with it some operation on the domains of a model, with respect to which the domains must be closed. The statement of completeness can be given precisely and proved for models meeting these closure conditions.

6 A demonstration of this type of completeness can be carried out along the lines of the author's recent paper, The completeness of the first order functional calculus, this Journal, vol. 14 (1949), pp. 159166Google Scholar.

7 Church, Alonzo, A formulation of the simple theory of types, this Journal, vol. 5 (1940), pp. 5668Google Scholar.

8 In addition to the notion of validity, the mappings V ϕ may be used to define the concept of the denotation of a wff A α containing no free occurrence of any variable. We first show (by induction) that if ϕ and ψ are two assignments which have the same value for every variable with a free occurrence in the wff B α, then V ϕ(B α) = V ψ(B α). Then the denotation of A α, is simply V ϕ(A α) for any ϕ. We also define the notion of satisfiability. If Γ is a set of wffs and ϕ an assignment with respect some model {D α) such that V ϕ(A ο) is T for every Aο in Γ, then we say that Γ is satisfiable with respect to the model {D α}. If Γ is satisfiable with respect to some model, we say simply that it is satisfiable.

9 Skolem, Thoralf, Über einige Grundlagenfragen der Mathematik, Skrifter utgitt av Det Norske Videnskaps-Akademi, I, no. 4 (1929), 49 ppGoogle Scholar.

10 Here we are identifying a set X of elements of D ι with the function (element of Dοι which maps every element of X into T and every other element of D ι into F.

11 Skolem, Thoralf, Über die Unmöglichkeit einer vollständigen Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems, Norsk matematisk forenings skrifter— series 2 no. 10 (1933), pp. 7382Google Scholar. And Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen, Fundamentamathematicae, vol. 23 (1934), pp. 150161Google Scholar.

12 A similar result for formulations of arithmetic within the first order functional calculus was established by Malcev, A., Untersuchungen aus dem Gebiete der mathematischen Logik, Recueil mathématique, n.s. vol. 1 (1936), pp. 323336Google Scholar. Malcev's method of proof bears a certain resemblance to the method used above. I am indebted to Professor Church for bringing this paper to my attention. (Added February 14, 1950.)

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