# 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. Robinsonβs 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 $V\beta \x81\u2019(S)$ defined by the conditions

$${V}_{0}\beta \x81\u2019(\mathrm{\pi \x9d\x90\x92})=\mathrm{\pi \x9d\x90\x92}$$ |

$${V}_{n+1}\beta \x81\u2019(\mathrm{\pi \x9d\x90\x92})={V}_{n}\beta \x81\u2019(\mathrm{\pi \x9d\x90\x92})\beta \x88\u037a{2}^{{V}_{n}\beta \x81\u2019(\mathrm{\pi \x9d\x90\x92})}$$ |

$$V\beta \x81\u2019(\mathrm{\pi \x9d\x90\x92})=\underset{n\beta \x88\x88\mathrm{\beta \x84\x95}}{\beta \x8b\x83}{V}_{n}\beta \x81\u2019(\mathrm{\pi \x9d\x90\x92})$$ |

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 $V\beta \x81\u2019(R)$.

The working view of nonstandard analysis is a set $*R$ and a mapping

$$*:V(\mathrm{\beta \x84\x9d})\beta \x86\x92V(*\mathrm{\beta \x84\x9d})$$ |

which satisfies some additional properties. $*\mathrm{\beta \x84\x9d}$ is of course embedded in $\mathrm{\beta \x84\x9d}$.

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:

$$\beta \x88\x80x\beta \x88\x88A,\mathrm{\Xi \xa6}\beta \x81\u2019(x,{\mathrm{\Xi \pm}}_{1},\mathrm{\beta \x80\xa6},{\mathrm{\Xi \pm}}_{n})$$ |

$$\beta \x88\x83x\beta \x88\x88A,\mathrm{\Xi \xa6}\beta \x81\u2019(x,{\mathrm{\Xi \pm}}_{1},\mathrm{\beta \x80\xa6},{\mathrm{\Xi \pm}}_{n})$$ |

For example, the formula

$$\beta \x88\x80x\beta \x88\x88A,\beta \x88\x83y\beta \x88\x88{2}^{B},x\beta \x88\x88y$$ |

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,

$$\beta \x88\x80x\beta \x88\x88A,\beta \x88\x83y,x\beta \x88\x88y$$ |

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 $V\beta \x81\u2019(R)$. $*A$ itself is internal if $A$ belongs to $V\beta \x81\u2019(R)$.

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

Transfer principle: For any formula $P\beta \x81\u2019({x}_{1},\mathrm{\beta \x80\xa6},{x}_{n})$ with bounded quantification and with free variables^{} ${x}_{1},\mathrm{\beta \x80\xa6},{x}_{n}$, and for any elements ${A}_{1},\mathrm{\beta \x80\xa6},{A}_{n}$ of $V\beta \x81\u2019(R)$, the following equivalence holds:
:

$$P({A}_{1},\mathrm{\beta \x80\xa6},{A}_{n})\beta \x87\x94P(*{A}_{1},\mathrm{\beta \x80\xa6},*{A}_{n})$$ |

Countable^{} saturation: If $A_{k}{}_{k}$ is a decreasing sequence of nonempty internal sets, with $k$ ranging over the natural numbers, then
:

$$\underset{k}{\beta \x8b\x82}{A}_{k}\beta \x89\mathrm{\beta \x88\x85}$$ |

One can show using ultraproducts^{} that such a map * exists. Elements of $V\beta \x81\u2019(R)$ 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

$${A}_{k}=\{k\beta \x88\x88*\mathrm{\beta \x84\x95}:k\beta \x89\u20afn\}$$ |

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

$$r\beta \x89\x85s\beta \x87\x94\beta \x88\x80\mathrm{\Xi \u0388}\beta \x88\x88{\mathrm{\beta \x84\x9d}}^{+},|r-s|\beta \x89\u20ac\mathrm{\Xi \u0388}$$ |

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 $\frac{1}{n}$ 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 http://en.wikipedia.org/wiki/Nonstandard_analysisNon-standard analysis as of December 19, 2006.

Title | non-standard analysis |
---|---|

Canonical name | NonstandardAnalysis |

Date of creation | 2013-03-22 16:28:26 |

Last modified on | 2013-03-22 16:28:26 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 13 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 03H05 |

Synonym | nonstandard analysis |