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Let “the hypothesis of similarity” be the hypothesis that all infinite classes are similar (have the same power or cardinal number). It will be shown that the system of logic of Whitehead and Russell's Principia mathematica (without the axiom of reducibility) is consistent when to its axioms are added all the following:
(1). The hypothesis of similarity.
(2). The axiom of infinity.
(3). The axiom of choice.
(4). The contradictory of the axiom of reducibility.
If (2) – (4) but not (1) are employed in Principia, it must therefore be the case that Cantor's theorem is not deducible, since it contradicts (1). If (2) and (3) but not (1) and (4) are employed, then a proof of Cantor's theorem must require the use of some sort of reducibility principle.
The first step in the required consistency proof is to modify in certain respects the system S of a previous paper. Immediately after definition 3.5 the following definition is to be added:
3.5.1. If a is of the form 
(b,c), where x is an S-variable and where b and c are S-constants neither of which is of higher order than that of x, then a is an S-proposition.
