Definition·D035
Cardinality of a finite set
The number of elements: the unique natural the set matches.
In words
The cardinality of a finite set is the unique natural number it is equinumerous with.
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
Uniqueness of this
is L23. Examples:
;
via the bijection
; and
for every natural
itself, via the identity. The basic counting laws: disjoint unions add (L26), subsets and images stay finite, and a partition into
classes of size
makes
(L27). Lagrange's theorem is a cardinality statement through and through.
Used by
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