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non-standard analysis (Definition)

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

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

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

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

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

$\displaystyle *: 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:

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

For example, the formula

$\displaystyle \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,
$\displaystyle \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: :

$\displaystyle 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 :

$\displaystyle \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

$\displaystyle 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

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

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.



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Cross-references: Wikipedia, adapted, bound, subsets, ideal, ring, subring, integer, intersection, superset, map, ultraproducts, decreasing, countable, equivalence, free variables, identity, extension, variable, restricted, range, occur in, quantifiers, bounded, formula, definitions, state, properties, mapping, topological vector spaces, metrizable, metric spaces, separable, isomorphic, contains, structures, sequence, union, power set, operation, numbers, Calculus, hyperreal, real numbers, satisfies, field, non-archimedean, natural number, absolute value, ordered field, infinitesimal, branch
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This is version 10 of non-standard analysis, born on 2006-12-19, modified 2007-11-18.
Object id is 8638, canonical name is NonStandardAnalysis.
Accessed 1428 times total.

Classification:
AMS MSC: 03H05 (Mathematical logic and foundations :: Nonstandard models :: Nonstandard models in mathematics)
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