Definition·D049
Domination
One set is dominated by another when it injects into it: a size comparison without counting.
For sets
, say
is dominated by
, written
, when there is an injection from
to
:
In words
A is dominated by B exactly when there is an injection from A to B.
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
Domination compares sizes without counting:
says a copy of
sits inside
with nothing identified. It is reflexive (the identity injects
into
) and transitive (composition of injections, (ii)). Its antisymmetry up to size, that
and
force
, is exactly the Cantor-Schröder-Bernstein theorem. By Cantor's theorem the ordering has no top:
always holds (via
), but
never does. Taking
gives countability.