Leon Henkin

Completeness in the Theory of Types

The first order functional calculus was proved complete by Godel 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 Godel 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 Godels 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.

The Completeness of the First-Order Functional Calculus

Although several proofs have been published showing the completeness of the propositional calculus (cf. Quine (1)), for the first-order functional calculus only the original completeness proof of Godel (2) and a variant due to Hilbert and Bernays have appeared. Aside from novelty and the fact that it requires less formal development of the system from the axioms, the new method of proof which is the subject of this paper possesses two advantages. In the first place an important property of formal systems which is associated with completeness can now be generalized to systems containing a non-denumerable infinity of primitive symbols. While this is not of especial interest when formal systems are considered as logics —i.e., as means for analyzing the structure of languages—it leads to interesting applications in the field of abstract algebra. In the second place the proof suggests a new approach to the problem of completeness for functional calculi of higher order. Both of these matters will be taken up in future papers. The system with which we shall deal here will contain as primitive symbols and certain sets of symbols as follows: (i) propositional symbols (some of which may be classed as variables , others as constants ), and among which the symbol “ f ” above is to be included as a constant; (ii) for each number n = 1, 2, … a set of functional symbols of degree n (which again may be separated into variables and constants ); and (iii) individual symbols among which variables must be distinguished from constants . The set of variables must be infinite.

Cylindric set algebras

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An extension of the Craig-Lyndon interpolation theorem

In a work widely quoted and applied, 3 Craig has shown that if A and C are any formulas of predicate logic such that A├C, then there is a formula B such that (i) A├B and B├C, and (ii) each predicate symbol occurring in B occurs both in A and in C . 4 If, in this theorem, we replace the syntactic notion of derivability, ├, by the semantical notion of consequence, ╞, the resulting proposition is of course equally valid, for by the (strong) completeness theorem of predicate logic 5 the relations ├ and ╞ coincide in extension.

The Logic of Equality

(1977). The Logic of Equality. The American Mathematical Monthly: Vol. 84, No. 8, pp. 597-612.

The Discovery of My Completeness Proofs

§1. Introduction . This paper deals with aspects of my doctoral dissertation which contributed to the early development of model theory. What was of use to later workers was less the results of my thesis, than the method by which I proved the completeness of first-order logic—a result established by Kurt Godel in his doctoral thesis 18 years before. The ideas that fed my discovery of this proof were mostly those I found in the teachings and writings of Alonzo Church. This may seem curious, as his work in logic, and his teaching, gave great emphasis to the constructive character of mathematical logic, while the model theory to which I contributed is filled with theorems about very large classes of mathematical structures, whose proofs often by-pass constructive methods. Another curious thing about my discovery of a new proof of Godels completeness theorem, is that it arrived in the midst of my efforts to prove an entirely different result. Such “accidental” discoveries arise in many parts of scientific work. Perhaps there are regularities in the conditions under which such “accidents” occur which would interest some historians, so I shall try to describe in some detail the accident which befell me. A mathematical discovery is an idea, or a complex of ideas, which have been found and set forth under certain circumstances. The process of discovery consists in selecting certain input ideas and somehow combining and transforming them to produce the new output ideas. The process that produces a particular discovery may thus be represented by a diagram such as one sees in many parts of science; a “black box” with lines coming in from the left to represent the input ideas, and lines going out to the right representing the output. To describe that discovery one must explain what occurs inside the box, i.e., how the outputs were obtained from the inputs.

On the Definition of `Formal Deduction'

1. The following remarks apply to many functional calculi, each of which can be variously axiomatized, but for clarity of exposition we shall confine our attention to one particular system E. This system is to have the usual primitive symbols2 and formation rules of the pure first-order functional calculus, and the following formal axiom schemata and formal rules of inference. Axiom schema i. Any tautologous wff (well-formed formula). Axiom schema 2. (a)A D B, where A is any wff, a and b are any individual variables, and B arises from A by replacing all free occurrences of a by free occurrences of b. Axiom schema 3. (a)(A D B) D (A D (a)B), where A and B are any wffs, and a is any individual variable not free in A. Rule of Modus Ponens: applies to wffs A and A D B, and yields B. Rule of Generalization: applies to a wff A and yields (a)A, where a is any individual variable. A formal proof in E is a finite column of wffs each of whose lines is a formal axiom or arises from two preceding lines by the Rule of Modus Ponens or arises from a single preceding line by the Rule of Generalization. A formal theorem of E is a wff which occurs as the last line of some formal proof. It is customary to define in terms of the formal axioms and rules, for each set r of wffs and for each wff A, a certain class D(F, A) of formal objects (usually columns of wffs) called formal deductions of A from assumptions r, and then to define the relation F by the stipulation that rFA if and only if D(r, A) is non-empty. It is desirable that the definition of D(r, A) be so formulated that the following conditions3 obtain. (i) There is an effective test to decide, for each decidable set r and

A Generalization of the Concept of

The concepts of ω-consistency and ω-completeness are closely related. The former concept has been generalized to notions of Γ- consistency and strong Γ- consistency , which are applicable not only to formal systems of number theory, but to all functional calculi containing individual constants; and in this general setting the semantical significance of these concepts has been studied. In the present work we carry out an analogous generalization for the concept of ω -completeness. Suppose, then, that F is an applied functional calculus, and that Γ is a non-empty set of individual constants of F . We say that F is Γ- complete if, whenever B(x) is a formula (containing the single free individual variable x ) such that ⊦ B (α) for every α in Γ, then also ⊦ (x)B(x) . In the paper “Γ-con” a sequence of increasingly strong concepts, Γ-consistency, n = 1,2, 3,…, was introduced; and it is possible in a formal way to define corresponding concepts of Γ n -completeness, as follows. We say that F is Γ n - complete if, whenever B(x 1 ,…, x n ) is a formula (containing exactly n distinct free variables, namely x 1 …, x n ) such that ⊦ B ( α 1 ,…, α n ) for all α 1 ,…, α n in Γ, then also ⊦ ( X 1 )…( x n ) B ( x 1 ,…, x n ). However, unlike the situation encountered in the paper “Γ-con”, these definitions are not of interest – for the simple reason that F is Γ n -complete if and only if it is Γ-complete, as one easily sees.

\omega

Publisher Summary This chapter focuses on the nominalistic analysis of mathematical language. The chapter focuses on the foundational theories that are directed toward the extent to which these are susceptible of mathematical formulation and toward the solution of technical problems that may arise from such analyses. In particular, this has characterized the approach to modern efforts to obtain a nominalistic interpretation of mathematical language. While the nominalistic tradition in philosophy is very ancient, a specific concentration of interest in this viewpoint—as applied especially to the analysis of mathematical language—can be discerned in the work of the Polish school of logicians. The difficulties in providing a nominalistic interpretation for particular sentences are discussed in the chapter. It is desirable to investigate what possibilities are for a nominalist to provide a systematic interpretation for all the sentences of some well-defined mathematical language that is adequate for most mathematical discourse.

-Completeness

Publisher Summary This chapter discusses certain algebraic structures introduced and studied by Alfred Tarski and F. B. Thompson. It represents theorem for cylindrical algebras (c.a). Cylindrical algebras are closely connected with formalisms, models, and formal calculi but the basic definitions is given without any reference to the metamathematical concepts, so that the representation theorem for c.a.‘s may be regarded as a theorem of algebra and set theory. It is pointed out that this theorem may be regarded as a generalization of the representation theorem for Boolean algebras.