Definition·D026
The natural numbers
The smallest inductive set: zero, one, two, and nothing else.
A natural number is a member of
. We write
,
,
,
, with
the successor.
In words
x belongs to ω exactly when for every set J, if J is inductive then x belongs to J. A natural number is a member of ω.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Existence: Infinity provides an inductive set
, and Separation carves out
; the result does not depend on which
was used, by Extensionality, since the displayed condition already forces membership in every inductive set. That
is itself inductive, and contained in every inductive set, is L04. In the von Neumann coding each natural is the set of all smaller ones:
,
,
,
; this is what lets the order be plain membership. The inductive shape of the definition makes induction almost immediate, and the Peano axioms all hold.
Used by
Propose an edit1 published revision