I intend to show that for S1, S2, and S3 the class of true formulas cannot coincide with the class of theorems. This is to hold irrespective of the meaning assigned to “◊”. when only “∼”, “▪”, and the variables are interpreted in the customary manner.

The idea of the proof is extremely simple and can be illustrated by the following argument concerning S3. The formula ~(◊(p▪~p)⥽▪p▪~p) ∨ (◊(q▪~q)⥽▪q▪~q) is a theorem of S3. Suppose now that all S3-theorems are true. Then either ~(◊(p▪~p)⥽▪p▪~p) is true (for every p) or (◊(q▪~q)⥽▪q▪~q) is true (for every q). But none of these formulas is an S3-theorem. Then some true S3-formula is not an S3-theorem. Hence, if all S3-theorems are true, some true S3-formula is not an S3-theorem. Then the class of S3-theorems cannot be identical with the class of true S3-formulas.

In a previous paper, I proved the consistency of a non-finitary system of logic based on the theory of types, which was shown to contain the axiom of reducibility in a form which seemed not to interfere with the classical construction of real numbers. A form of the system containing a strong axiom of choice was also proved consistent.

It seems to me now that the real-number approach used in that paper, though valid, was not the most fruitful one. We can, on the lines therein suggested, prove the consistency of axioms closely resembling Tarski's twenty axioms for the real numbers; but this, from the standpoint of mathematical practice, is a pitifully small fragment of analysis. The consistency of a fairly strong set-theory can be proved, using the results of my previous paper, with little more difficulty than that of the Tarski axioms; this being the case, it would seem a saving in effort to derive the consistency of such a theory first, then to strengthen that theory (if possible) in such ways as can be shown to preserve consistency; and finally to derive from the system thus strengthened, if need be, a more usable real-number theory. The present paper is meant to achieve the first part of this program. The paragraphs of this paper are numbered consecutively with those of my previous paper, of which it is to be regarded as a continuation.