A "mediate cardinal" is defined in "Principia Mathematica" as a cardinal which is neither inductive nor reflexive and it is established *(124·61) that the multiplicative axiom implies the non-existence of mediate cardinals. The converse implication is not established, and there seems to be no reason to suppose it is true. The relation of the existence of mediate cardinals to the multiplicative axiom is therefore one-sided and offers a contrast to the mutual implications of the comparability of cardinals, the well-orderability of classes and the multiplicative axiom. In this paper it is proposed to investigate other classes of cardinals which are not Alephs, beyond the mediate cardinals of "Principia Mathematica," and instead of the one-sided implication between the multiplicative axiom and the non-existence of mediate cardinals to establish an equivalence between the axiom and the non-existence of certain cardinals which are not Alephs.
If we use the term "comparable" in such a sense that μ is comparable with ν when μ is greater than, equal to, or less than ν, a mediate cardinal in "Principia Mathematica" is a cardinal comparable with the inductive cardinals but not with A0 (A has been substituted for the Hebrew Character Aleph usually employed). We wish to discuss the nature of those cardinals (if there are any such) which are comparable with Alephs less than Aξ, but not with Aξ, for different values of ξ. In "Principia Mathematica" it is found that a part of the multiplicative axiom is sufficient to imply the non-existence of the non-inductive non-reflexive cardinals and instead of the proposition
Mult Ax . ⊃ . NC med = Λ
there is the proposition *124·56
A0 ∈ NC mult . ⊃ . NC med = Λ.
It seems worth while to pursue the same course with the cardinals to be considered in this paper.