Definition·D034
Finite set
A set that matches some natural number.
with
from D033 and
the naturals. A set that is not finite is infinite.
In words
A set is finite exactly when it is equinumerous with some natural number.
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
In the von Neumann coding,
is itself an
-element set, so "counting
" literally means matching
against a numeral:
(the empty function), and a three-element set matches
. The matching number is unique (L23, via the pigeonhole principle), which makes cardinality well defined. Finiteness is robust: it passes to subsets and images, and disjoint unions add sizes (L26).
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