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# non-standard analysis

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.

## Mathematics Subject Classification

03H05*no label found*

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