³ This mutual deducibility was pointed out to me by Professor Quine, who also made many other very helpful criticisms and suggestions. I am also indebted to him for his kind encouragement.

⁴ It seems that we might, without using the powers of relations, have defined the matrix ‘x N y’ as follows:

But it is difficult to derive the desired theorems with this definition.

⁵ At first, I defined ‘x < y’ as ‘x ε Nn ▪ (⋆Є | Є) (x, y)’. The present version D41 is G. D. W. Berry's. He points out that, under my former definition, the theorems †665–667 are in themselves consistent with there being only a finite number of natural numbers plus one object y not a natural number but such that (x) (x ε Nn ▪ ⊃. x < y)

I am also indebted to Berry for numerous less striking corrections and suggestions.

⁶ We might have reduced all the defined symbols in our axioms to statement connectives and quantifiers plus ‘ε’, and merely set down formal truth-conditions for the schema in terms of our arithmetical model—thereby indicating the consistency of our rather weak axioms.