The Principles of Mathematics (1903)

§ 498

Numbers, also, are a type lying outside the above series, and presenting certain difficulties, owing to the fact that every number selects certain objects out of every other type of ranges, namely those ranges which have the given number of members. This renders the obvious definition of 0 erroneous; for every type of range will have its own null-range, which will be a member of 0 considered as a range of ranges, so that we cannot say that 0 is the range whose only member is the null-range. Also numbers require a consideration of the totality of types and ranges; and in this consideration there may be difficulties.(§ 498 ¶ 1)

Since all ranges have numbers, ranges are a range; consequently xx is sometimes significant, and in these cases its denial is also significant. Consequently there is a range w of ranges for which xx is false: thus the Contradiction proves that this range w does not belong to the range of significance of xx. We may observe that xx can only be significant when x is of a type of infinite order, since, in xu, u must always be of a type higher by one than x; but the range of all ranges is of course of a type of infinite order.(§ 498 ¶ 2)

Since numbers are a type, the propositional function x is not a u, where u is a range of numbers, must mean x is a number which is not a u; unless, indeed, to escape this somewhat paradoxical result, we say that, although numbers are a type in regard to certain propositions, they are not a type in regard to such propositions as u is contained in v or x is a u. Such a view is perfectly tenable, though it leads to complications of which it is hard to see the end.(§ 498 ¶ 3)

That propositions are a type results from the fact--if it be a fact--that only propositions can significantly be said to be true or false. Certainly true propositions appear to form a type, since they alone are asserted (cf. Appendix A. § 479). But if so, the number of propositions is as great as that of all objects absolutely, since every object is identical with itself, and x is identical with x has a one-one relation to x. In this there are, however, two difficulties. First, what we called the propositional concept appears to be always an individual; consequently there should be no more propositions than individuals. Secondly, if it is possible as it seems to be, to form ranges of propositions, there must be more such ranges than there are propositions, although such ranges are only some among objects (cf. § 343). These two difficulties are very serious, and demand a full discussion.(¶ 498 ¶ 4)