One proof that a certain set does not have this property lists the elements and changes the ith digit of the ith entry for all i; that is Cantor's diagonalization argument.
If a theory with this property is satisfiable then it has a model of every infinite cardinality by the Lowenheim-Skolem theorem. An algebraic closure either has the same cardinality as the initial field or has this property. The sets of a sigma algebra are closed under this type of union. Zorn's lemma is frequently used rather than induction on sets without this property. Sets with cardinality (*) Aleph-Naught have this property. One proof that a certain set does not have this property lists the elements and changes the ith digit of the ith entry for all i; that is Cantor's diagonalization argument. For 10 points, name this property of a set which is not possessed by the reals, but possessed by the rationals and integers, defined as having the same cardinality as the natural numbers.
If a theory with this property is satisfiable then it has a model of every infinite cardinality by the Lowenheim-Skolem theorem.
countability [or countably infinite; accept word forms; prompt on infinite]