Nonstandard analysis is a rich formalization of analysis that uses a certain explicit notions of infinitesimal objects. In fact, not only infinitesimal but also infinitely large can be accomodated (and must be). Moreover, not only the field of real numbers, but more general algebraic structures can be extended, essentially via a construction of ultraproducts; also general sets can be extended to contain nonstandard elements (see internal set).
See also nonstandard analysis in topology.
At its beginning, infinitesimal calculus was developed nonrigorously, though many interesting arguments and formal manipulations were found. Cauchy and Weierstraß introduced the $\epsilon$-$\delta$ approach, which enabled modern rigorous analysis, but sometimes this method is cumbersome. For example, sometimes one needs to work with several infinitesimal levels or kinds of continuity in the same problem, and finding estimates may be very cumbersome. One would like to introduce infinitesimal quantities as additional elements of the sets of usual (‘standard’) quantities. Several related rigorous frameworks appeared under the name of nonstandard analysis, since the first such discovered by Abraham Robinson. Most often, approaches using ultrafilters, certain classes called internal sets and using topos theory enable the foundation of nonstandard analysis. Many properties and theorems from classical analysis imply new statements of nonstandard analysis; the mechanism is the so-called transfer principle, which can be required axiomatically without respect to a particular model of nonstandard analysis.
Assuming the axiom of choice (whose full strength is not necessary), there exists a free (= not containing finite subsets) ultrafilter $F$ on the set of natural numbers $\mathbb{N}=\{1,2,3,\ldots\}$, and such ultrafilters are in $1$–$1$ correspondence with finitely additive measures on $\mathbb{N}$ (using the algebra of all subsets) taking values in the two element set $\{0,1\}$.
Fix a free ultrafilter $F$ on $\mathbb{N}$, and consider the set of all sequences of real numbers, $\mathbb{R}^{\mathbb{N}}$. So an element in here is a sequence
of real numbers.
We write $f\sim_F g$ if the set $\{i\in\mathbb{N}|f(i)=g(i)\}$ belongs to $F$; these are precisely the sequences which are equal almost everywhere with respect to the associated measure. The relation $\sim_F$ is an [[equivalence relation] and ${}^*\mathbb{R} := \mathbb{R}^{\mathbb{N}}/{\sim_F}$ is a nonstandard extension of $\mathbb{R}$, whose elements are sometimes called hyperreal numbers.
(This is a special case of the ultraproduct construction in model theory. In fact, we could have started with an ultrafilter on any set, not just $\mathbb{N}$. Such more general ultraproducts are necessary in order to obtain more refined models of nonstandard analysis satisfying stronger “saturation” principles.)
Given $f\in \mathbb{R}^{\mathbb{N}}$, we write $f_F$ for its equivalence class in ${}^*\mathbb{R}$. In particular, given any real number $r\in \mathbb{R}$ the image $^* r :=(i\mapsto r)_F$ of the constant sequence
is an element in $^*\mathbb{R}$ and this gives an injection $*:\mathbb{R}\hookrightarrow {}^*\mathbb{R}$.
$^*\mathbb{R}$ is equipped with a linear ordering given by
which makes $*:\mathbb{R}\hookrightarrow {}^*\mathbb{R}$ a monotone function.
An element $\delta\in{}^*\mathbb{R}$ such that $^* 0\lt\delta$ is called a positive number .
An element $\delta$ such that for all positive $r\in \mathbb{R}$ we have $-{}^* r \lt \delta\lt{}^* r$ is called an infinitesimal number.
Unlike in the real numbers, positive infinitesimal numbers exist: for example the class $f_F$ where $f:n\mapsto 1/n$ is such and $g_F$ for $g:n\mapsto 1/n^2$ is a different one.
Let $n$ be a nonnegative integer and $u:\mathbb{R}^n\to\mathbb{R}$ a function. Then there is a nonstandard extension $^* u:({}^*\mathbb{R})^n\to{}^*\mathbb{R}$ of $f$; it is defined by
This is indeed an extension of $u$ in the sense that $^* u({}^*r_1,\ldots,{}^*r_n)={}^* r$ iff $u(r_1,\ldots,r_n)=r$. This way, the usual operations $+,\cdot$ and the absolute value $|\cdot|$ extend to $^*\mathbb{R}$; usually one denotes these and other standard operations on ${}*\mathbb{R}$ without putting $^*$ in front, writing simply e.g. $f_F+g_F$.
To extend division appropriately, we need a little bit more care as it is originally just partially defined, so we need an extension of the formalism to subsets of the real line. In particular there is a following definition of an extension $^* E\subset{}^*\mathbb{R}$ of a subset $E\subset\mathbb{R}$:
(For example, a number is positive, as defined earlier, if an only if it belongs to $^*\{r|r \gt 0\}$.) Then division is extended to a function $^*\mathbb{R} \times ^*\{r|r \ne 0\} \to ^*\mathbb{R}$. If $1/x$ is infinitesimal, then $x$ itself is infinite.
Conversely, an element $x\in{}^*\mathbb{R}$ is finite if $|x|\lt {}^* r$ for some $r\in\mathbb{R}$. Every finite element $x\in{}^*\mathbb{R}$ is infinitely close to a unique real number $q\in\mathbb{R}$ in the sense that $x-{}^*q$ is infinitesimal. We say that $q$ is the standard part of $x$ and is denoted by $q= st(x)$. Given a real number $r\in\mathbb{R}$, the subset $\mu(r)$ of all elements $x\in{}^*\mathbb{R}$ such that $st(x)=r$ is said to be the monad of the real number $r\in\mathbb{R}$. Monads should be thaught of as infinitesimal neighborhoods. An elementary fact: a subset $E\subset\mathbb{R}$ is open iff $\mu(r)\subset{}^*E$ for all $r\in E$; $E$ is closed iff $st(x)\in E$ for all finite $x\in{}^* E$; and $E$ is compact iff, for all $x\in{}^* E$, $x$ is finite and $st(x)\in E$.
In this model of nonstandard analysis, the transfer principle? is a corollary of a general theorem on ultraproducts due Jerzy Łoś. It can be stated in terms of a certain formal language $L(\mathbb{R})$ of the real numbers. We can also extend this model to ultrapowers of larger sets, not just $\mathbb{R}$ itself, with a corresponding extension of the language. In the limit where we reach an entire “universe” of mathematics, this leads to the topos-theoretic filterquotient and sheaf models below.
The ultrapower construction above can be performed in the general context of topos theory. From any topos $\mathcal{E}$ and any filter $\Phi$ of subterminal objects in $\mathcal{E}$, one can construct a topos $\mathcal{E}/\Phi$, the filterquotient? of $\mathcal{E}$ by $\Phi$. There is a logical functor $\mathcal{E} \to \mathcal{E}/\Phi$.
If $\mathcal{E} = Set / \mathbb{N}$, then any filter on $\mathbb{N}$ gives a filter of subterminals in $\mathcal{E}$, whose corresponding filterquotient corresponds to the above construction. The composite functor
might be written $^*(-)$. If $\Phi$ is an ultrafilter, then $(Set/\mathbb{N})/\Phi$ is a two-valued topos, whose internal logic is essentially that of the ultrafilter model described above. In particular, the global elements of $^*\mathbb{R}$, as an object of this topos, are precisely the “hyperreal numbers” described above.
In this context, the transfer principle? is the fact that the functor $^*(-)$ is both logical and conservative, and hence it both preserves and reflects the truth of formulas in the internal languages.
A different topos-theoretic construction is to consider the topos of sheaves on a category of filters. This topos models the internal set theory of Nelson, a more axiomatic approach to nonstandard analysis. References:
The Lebesgue measure on $\mathbf{R}^n$ extends to Loeb measure on ${}^\ast\mathbf{R}^n$. This may be used for probability theory and also for generalized functions.
The theory of generalized functions of Schwarz can be reproduced by nonstandard analysis:
Theorem. (Abraham Robinson) Every generalized function $f:\mathbf{R}\to\mathbf{R}$ can be represented as the integration of the product of test function with a nonstandard smooth function $\tilde{f}: {}^\ast\mathbf{R}\to{}^\ast \mathbf{R}$
There is also inutionistic version of nonstandard analysis approach to generalized functions as well as nonstandard approaches to Sato hyperfunctions (Sousa pinto), to Coulombeu distributions etc.
There are other ways of realizing the notion of infinitesimal number precisely, such as synthetic differential geometry and the surreal numbers. Neither seem to be very closely related to NSA—the techniques and flavor of each subject are quite different. However, some things can be said.
While the most common infinitesimals appearing in SDG are nilpotent, in contrast to those of NSA which are invertible, some models of SDG do contain invertible infinitesimals; see here.
Since the surreal numbers are the universally embedding ordered field, any field of hyperreals can be embedded in the surreals. However, such embeddings don’t seem very useful, since they don’t preserve any of the important structure of the hyperreals (such as the transfer principle).
Sergio Albeverio?, Jens Erik Fenstad, Raphael Hoegh-Krohn, Nonstandard methods in stochastic analysis and mathematical physics, Academic Press 1986 (there is also a Dover 2009 edition and a 1990 Russian translation)
Jerome Keisler?, Elementary calculus: an infinitesimal approach, online undergraduate textbook.
Abraham Robinson, Non-standard analysis, 1966
Wikipedia: nonstandard analysis, ultraproduct, hyperreal numbers, Abraham Robinson, constructive non-standard analysis, criticism of nonstandard analysis
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