nLab natural numbers structure

Contents

 Idea

Abstracting the von Neumann ordinals and the Zermelo ordinals, amongst other definitions of the natural numbers, in material set theory.

Definition

Given a set VV with an extensional relation \prec, a natural numbers structure is an element V\mathbb{N} \in V, an element 0V0 \in V, and a function s:VVs:V \to V such that

  • 00 \prec \mathbb{N}
  • for all xVx \in V, if xx \prec \mathbb{N}, then s(x)s(x) \prec \mathbb{N}
  • for all zVz \in V, if 0z0 \prec z and for all xVx \in V, if xzx \prec z, then s(x)zs(x) \prec z, then for all xVx \in V, xx \prec \mathbb{N} implies that xzx \prec z.

Foundational concerns

Usually in material set theory, 00 is defined to be the empty set \emptyset, \mathbb{N} is defined to be the von Neumann ordinal ω\omega, and ss is defined to be the operation xx{x}x \mapsto x \cup \{x\}. However, there are alternative options, such as using the Zermelo ordinals, where the operation ss is given by x{x}x \mapsto \{x\}, amongst others.

One could avoid having to choose amongst the definitions and abstract it all by adding primitive constants \mathbb{N} and 00 and primitive unary operation ss to the set theory, satisfying the axiom

  • 00 \in \mathbb{N}; for all sets xx, if xx \in \mathbb{N}, then s(x)s(x) \in \mathbb{N}; and for all sets zz, if 0z0 \in z and for all sets xx, if xzx \in z, then s(x)zs(x) \in z, then for all sets xx, xx \in \mathbb{N} implies that xzx \in z.

See also

Last revised on December 12, 2022 at 20:00:08. See the history of this page for a list of all contributions to it.