The proposition that a decreasing sequence of ordinals necessarily terminates has been given a new, and perhaps unexpected, importance by the rôle which it plays in Gentzen's proof of the freedom from contradiction of the “reine Zahlentheorie.” Gödel's construction of non-demonstrable propositions and the establishment of the impossibility of a proof of freedom from contradiction, within the framework of a certain type of formal system, showed that a proof of freedom from contradiction could be found only by transcending the axioms and proof processes of that formal system. Gentzen's proof succeeds by utilising transfinite induction to prove that certain sequences of reduction processes, enumerated by ordinals less than ε (the first ordinal to satisfy ε = ὡ) are finite. Were it possible to prove the restricted ordinal theorem, that a descending sequence of ordinals, less than ε, is finite, in Gentzen's “reine Zahlentheorie,” then it would be possible to determine a contradiction in that number system. In his paper, Gentzen proves the theorem of transfinite induction, which he requires, by an intuitive argument. There is also a method of reducing transfinite induction, for ordinals less than ε, to a number-theoretic principle given by Hilbert and Bernays, and a similar method by Ackermann. None of these proofs of transfinite induction is finitist.

One of the preeminent problems confronting logicians is that of constructing a system of logic which will be adequate for mathematics. By a system's being adequate for mathematics, we mean that all mathematical theorems in general use can be deduced within the system. Several distinct logical systems, all having this end in view, have been proposed. Among these perhaps the best known are the systems referred to as “Principia Mathematica” and “set theory.” In both of these systems (we refer to the revised and simplified versions) there is a nucleus of propositions which can be derived by using only the axioms and rules of the restricted predicate calculus. However, if anything like adequacy for mathematics is to be expected, additional primitives and axioms must be added to the restricted predicate calculus. It is in their treatment of the additional primitive ε, denoting class or set membership, that the above-mentioned systems differ.

In addition to these two, a third and a stronger system has been proposed by W. V. Quine in his paper New foundations for mathematical logic. It is with this system of Quine's that our work is concerned and of which we now give a brief description.

A previous paper proposed a finitary system of logic which was surmised to be basic in the sense that every finitary system of logic could be defined within it. In the present paper this anticipated result, or rather its syntactical counterpart, is definitely established as Theorem VI. The calculus K is a formalization of the basic logic. Its main features were described in the previous paper, acquaintance with which is presupposed here. Attention is called to the fact that from now on the class Y of atomic U-expressions will be assumed to be finite instead of infinite. This is a relatively minor change and does not affect the previously established results. The account of the nature of calculi, as given in the other paper, is to be made more precise by stating that a calculus is any class of expressions having as a Gödel representation a recursively enumerable class of integers. A system of logic is regarded as being finitary if it is capable of formalization by means of an appropriate calculus.

A logical calculus will be presented which not only is a formulation of a “basic logic” in the sense of the writer's previous papers, but which has the additional property that no weaker calculus can be a formulation of a basic logic. A sort of minimum logical calculus is thus attained, which has nothing superfluous about it for achieving the purpose for which it is designed.

In the case of some logical calculi the question can arise as to whether certain of the postulates are really logically valid and necessary. Sometimes a test is available, such as the truth-table test, enabling us to distinguish between logically valid sentences and others, but often no such test is available, especially where quantifiers are involved. Is or is not the axiom of infinity, for example, to be regarded as logically valid? Or is the principle of double negation really acceptable, even though it satisfies the truth-table test?

We say that a system S of sentential calculus is an extension of a system R, if R and S have the same class of (meaningful) sentences, and every provable sentence of R is also provable in S. If the two classes of provable sentences do not coincide, we call S a proper extension of R.

By a complete system of sentential calculus is meant one which is itself consistent, but has no consistent proper extensions. Thus one cannot add a new independent primitive sentence to a complete system without obtaining an inconsistent system. The usual two-valued sentential calculus is complete in the sense defined.

It has been shown by Lindenbaum that every incomplete system of sentential calculus possesses at least one complete extension. In this paper we shall examine how many complete extensions there are of some of the Lewis systems of sentential calculus. We shall show that there is only one complete extension of S4 (and hence also of S5, which is an extension of S4), and that there are infinitely many complete extensions of S2 (and hence also of S1, since S2 is an extension of S1). We leave open the question how many complete extensions there are of S3.