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.

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.