Successors of natural numbers

Given a natural number nn, the successor n +n^+ of nn is simply n+1n + 1. In a topos, a natural numbers object \mathbb{N} is equipped with a successor morphism \mathbb{N}\to\mathbb{N}, which, together with its zero element 11\to \mathbb{N}, is used to characterize its abstract universal property of recursion.

Successors in well-orderings and cardinals

More generally, in any well-ordered set SS, the successor w +w^+ of an element ww is the least element of SS which is (strictly) greater than ww (if such an element exists). If SS has no maximal element, then the successor map ww +w \mapsto w^+ is always defined; it is sometimes used to make recursive definitions. We say that an element of a well-ordered set is a successor if it is the successor of something.

This notion is sometimes also used for some well-ordered proper classes, for example for the classes Ord\mathbf{Ord} of ordinal numbers and CardCard of cardinal numbers. (The latter is only well-ordered if we assume the axiom of choice.) Thus the successor ordinal of an ordinal α\alpha is the least ordinal greater than α\alpha, which, if we use the von Neumann definition of ordinals, is α{α}\alpha \cup \{\alpha\}. Similarly, the successor cardinal of a cardinal κ\kappa is the least cardinal greater than κ\kappa. We speak of “a successor cardinal” or “a successor cardinal” to mean a cardinal which is the successor of some other ordinal or cardinal, respectively.

Note that if we identify a cardinal number with the least ordinal having that cardinality, as is common in material set theory, then the successor ordinal and the successor cardinal of that cardinal are different. For instance, the successor ordinal of ω= 0\omega = \aleph_0 is ω+1\omega+1, whereas its successor cardinal is 1\aleph_1.

For definitions by transfinite recursion?, one usually specifies the value at 00, the rule for recursion along the successor map, and a separate rule of recursion for the limiting ordinals (infinite ordinals which are not successors). (For example, the von Neumann hierarchy of well-founded pure sets is defined in that way.) One can (and in constructive mathematics must) also handle all three cases at once, and the successor function is used there as well.

Revised on November 16, 2012 18:36:57 by Mike Shulman (