5.252

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Only in this way is the progress from term to term in a formal series possible (from type to type in the hierarchy of Russell and Whitehead). (Russell and Whitehead have not admitted the possibility of this progress but have made use of it all the same.)

5.2521    The repeated application of an operation to its own result I call its successive application ("O' O' O' a" is the result of the threefold successive application of "O'  xi " to "a").

In a similar sense I speak of the successive application of several operations to a number of propositions.

5.2522    The general term of the formal series a,  O' a, O' O' a, ... I write thus: "[a, x, O' x]". This expression in brackets is a variable. The first term of the expression is the beginning of the formal series, the second the form of an arbitrary term x of the series, and the third the form of that term of the series which immediately follows x.

5.2523    The concept of the successive application of an operation is equivalent to the concept "and so on".