Non-standard model of arithmetic

In Model Theory, a nonstandard model of arithmetic (or, equivalently, a nonstandard model of number theory) is a model of all of number theory (i.e. the true statements in number theory are true in the new model), but with a larger underlying set (or universe). Both countable and uncountable nonstandard models of arithmetic exist. It is important to note that while the nonstandard model satisfies all of standard number theory, it also satisfies new sentences (e.g. one could construct a model of number theory in which the Twin Prime Conjecture holds).