Quote: the natural numbers are the posterity of 0 w.r.t. immediate predecessor; i.e., all larger numbers, by hereditary; includes Peano's 0 and induction

Quotation

A property is said to be "hereditary" in the natural-number series if, whenever it belongs to a number n, it also belongs to n+1, the successor of n. ... [p. 22] We will define the "posterity" of a given natural number with respect to the relation "immediate predecessor" (which is the converse of "successor") as all those terms that belong to every hereditary class to which the given number belongs [i.e., all larger numbers]. ... We now lay down the following definition :-- The "natural numbers" are the posterity of 0 with respect to the relation "immediate predecessor". ... As a result of this definition, two of [Peano's] primitive propositions--namely, the one asserting that 0 is a number and the one asserting mathematical induction--become unnecessary, since they result from the definition. ... [p. 25, due to Frege, Begriffsschrift 1879. Russell may have the first reader]   Google-1   Google-2

Published before 1923

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