"There are three main tools in nonstandard analysis. One is the transference principle, which roughly states that the same assertions of the formal language are true in the standard universe as in the nonstandard universe. It is typically used by proving a desired result in the nonstandard universe, and then, noting that the result is expressible in the language, concluding that it holds in the standard universe as well. Another technique is concurrence. This is a logical technique that guarantees that the extended structure contains all possible completions, compactifications, and so forth. The third technique is internality. A set s of elements of the nonstandard universe is internal if s itself is an element of the nonstandard universe; otherwiise, s is external. A surprislingly useful method of proof is one by reductio ad absurdum in which the contradiction is that some set one knows to be external would in fact be internal under the assumption being refuted."