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Finite
A set 
is finite if there exists a natural number
and a bijection
from 
to 
. Note that we are using the set theoretic definition of natural number, under which the natural number 
equals the set
. If there exists such an 
, then it is unique, and we call 
the cardinality of 
.
Equivalently, a set 
is finite if and only if there is no bijection between 
and any proper subset
of 
.