Infinite sets may be countable or uncountable. The power set of an infinite (either countable or uncountable) set is always uncountable. Countable nouns generally have singular and plural forms. Theorem&colon; Any finite union of countable sets is countable. Theorem&colon; The Cartesian product of finitely many countable sets is countable. The union of countably many countable sets is countable. A topological space with a countable dense subset is called separable. The classification is in terms of countable ordinal numbers. It makes use of countable nonstandard models of Peano arithmetic. Theorem&colon; Every subset of a countable set is countable. Discrete probability theory needs only at most countable sample spaces formula 2. There is only one possible order type of a countable nonstandard model. It is natural to ask whether a countable nonstandard model can be explicitly constructed. For inductions and recursions of countable length, the weaker axiom of dependent choice is sufficient. Any subset of a countable set is countable.