Non-standard analysis

Non-standard analysis is the usage of Model theory to study analysis. Since studying the [saturated model]? of a theory is easier then studying other models, non-standard analysis studies the [saturated model]? of theories with many symbols thrown in to make sure results are applicable.

One kind of elements that are in the [saturated model]? are infinitesimals. Since it is consistent for a real number to be smaller then any finite subset of {1/n| n natural}, there is a non-standard real number smaller then all of them. In fact, there is a whole ideal of non-standard real numbers. If we start from the rationals, rather then the real numbers, and divide the ring of non-standard finite rational numbers by the ideal of the infinitesimal rational numbers, we get a field (because it is a maximal ideal) -- the field of real numbers. This kind of easy ways to get results which are hard work in classic, epsilon-delta analysis is typical. For example, proving that the composition of continuous functions is continuous is much easier.

There are not many results proven first with non-standard analysis. One of them is the fact that every [polynomially compact linear operator]? on a Hilbert space has an invariant subspace, proven 5 years before classic functional analysis techniques were developed that deal with such problems.