The consistency of the ramified Principia

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Abstract
When the axioms of infinity and choice are added to the system of logic of the second edition of Whitehead and Russell's Principia mathematica, and when, as in the second edition, the axiom of reducibility is omitted (so that the ramified nature of Russell's theory of types, especially his distinction between “orders,” is not in effect obliterated), the resulting system will hereafter be referred to as “the ramified Principia.” It will be shown that a certain system S, slightly stronger than the ramified Principia, is consistent, so that the ramified Principia itself is consistent. From Gödel's theorem it will then follow that the sort of mathematical induction employed in the consistency proof of S cannot be adequately handled within S or within the ramified Principia. The method of proving the consistency of S will be roughly as follows: A consistent non-constructive system S′ will be defined by means of induction with respect to a serial well-ordering of all the propositions of S. It will then be shown that every true proposition of S is a true proposition of S′, so that S must also be consistent. 1. Preliminary conventions. 1.1. Definition. By a “symbol” is to be understood any typographical expression which does not consist of a horizontal collinear sequence of expressions. Expressions formed by adding indices to either of the two upper or two lower corners of a symbol will also be considered symbols.