Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field $F$ is infinitesimal if and only if its absolute value is smaller than any element of $F$ of the form $\frac{1}{n}$ , for $n$ a natural number. Ordered fields that have infinitesimal elements are also called non-Archimedean. More generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. A field which satisfies the transfer principle for real numbers is a hyperreal field, and non-standard real analysis uses these fields as non-standard models of the real numbers.

Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. Robinson's original approach was based on these non-standard models of the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966 and is still in print.

Several technical issues must be addressed to develop a calculus of infinitesimals. For example, it is not enough to construct an ordered field with infinitesimals. See the article on hyperreal numbers for a discussion of some of the relevant ideas.

Given any set $S$ , the superstructure over a set $S$ is the set $V(S)$ defined by the conditions

$$V_0(\mathbf{S}) = \mathbf{S}$$

$$V_{n+1}(\mathbf{S}) =V_{n}(\mathbf{S}) \cup 2^{V_{n}(\mathbf{S})}$$

$$V(\mathbf{S}) = \bigcup_{n \in \mathbb{N}} V_{n}(\mathbf{S})$$

Thus the superstructure over $S$ is obtained by starting from $S$ and iterating the operation of adjoining the power set of $S$ and taking the union of the resulting sequence. The superstructure over the real numbers includes a wealth of mathematical structures: For instance, it contains isomorphic copies of all separable metric spaces and metrizable topological vector spaces. Virtually all of mathematics that interests an analyst goes on within $V(R)$ .

The working view of nonstandard analysis is a set $*R$ and a mapping $$ *: V(\mathbb{R}) \rightarrow V(*\mathbb{R}) $$ which satisfies some additional properties. $*\mathbb{R}$ is of course embedded in $\mathbb{R}$ .

To formulate these principles we state first some definitions: A formula has bounded quantification if and only if the only quantifiers which occur in the formula have range restricted over sets, that is are all of the form:

$$ \forall x \in A, \Phi(x, \alpha_1, \ldots, \alpha_n) $$ $$ \exists x \in A, \Phi(x, \alpha_1, \ldots, \alpha_n) $$

For example, the formula

$$ \forall x \in A, \ \exists y \in 2^B, \ x \in y $$ has bounded quantification, the universally quantified variable $x$ ranges over $A$ , the existentially quantified variable $y$ ranges over the powerset of $B$ . On the other hand, $$ \forall x \in A, \ \exists y, \ x \in y $$ does not have bounded quantification because the quantification of $y$ is unrestricted.

A set $x$ is internal if and only if x is an element of $*A$ for some element $A$ of $V(R)$ . $*A$ itself is internal if $A$ belongs to $V(R)$ .

We now formulate the basic logical framework of nonstandard analysis: Extension principle: The mapping $*$ is the identity on $R$ .

Transfer principle: For any formula $P(x_1, \ldots, x_n)$ with bounded quantification and with free variables $x_1, \ldots, x_n$ , and for any elements $A_1, \ldots, A_n$ of $V(R)$ , the following equivalence holds: :$$P(A_1, \ldots, A_n) \iff P(*A_1, \ldots, *A_n) $$

Countable saturation: If ${A_k}_k$ is a decreasing sequence of nonempty internal sets, with $k$ ranging over the natural numbers, then :$$\bigcap_k A_k \neq \emptyset $$

One can show using ultraproducts that such a map * exists. Elements of $V(R)$ are called standard. Elements of $*R$ are called hyperreal numbers.

The symbol $*N$ denotes the nonstandard natural numbers. By the extension principle, this is a superset of $N$ . The set $*N - N$ is not empty. To see this, apply countable saturation to the sequence of internal sets

$$ A_k = \{k \in *\mathbb{N}: k \geq n\} $$

The sequence ${A_k}_k$ is in $N$ has a non-empty intersection, proving the result.

We begin with some definitions: Hyperreals $r$ , $s$ are infinitely close if and only if

$$ r \cong s \iff \forall \theta \in \mathbb{R}^+, \ |r - s| \leq \theta$$

A hyperreal $r$ is infinitesimal if and only if it is infinitely close to 0. $r$ is limited or bounded if and only if its absolute value is dominated by a standard integer. The bounded hyperreals form a subring of $*R$ containing the reals. In this ring, the infinitesimal hyperreals are an ideal. For example, if $n$ is an element of $*N - N$ , then ${1 \over n}$ is an infinitesimal.

The set of bounded hyperreals or the set of infinitesimal hyperreals are external subsets of $V(*R)$ ; what this means in practice is that bounded quantification, where the bound is an internal set, never ranges over these sets.

This entry was adapted from the Wikipedia article Non-standard analysis as of December 19, 2006.