Thehistory of calculus is fraught with philosophical debates about the meaning and logical validity offluxions orinfinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus usinglimits rather than infinitesimals.Nonstandard analysis^[1][2][3] instead reformulates the calculus using a logically rigorous notion ofinfinitesimal numbers.

Nonstandard analysis originated in the early 1960s by the mathematicianAbraham Robinson.^[4][5] He wrote:

... the idea of infinitely small orinfinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection ... that the distance between two distinct real numbers cannot be infinitely small,Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which wereto possess the same properties as the latter.

Robinson argued that thislaw of continuity of Leibniz's is a precursor of thetransfer principle. Robinson continued:

However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.^[6]

Robinson continues:

... Leibniz's ideas can be fully vindicated and ... they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporarymodel theory.

In 1973,intuitionistArend Heyting praised nonstandard analysis as "a standard model of important mathematical research".^[7]

A non-zero element of anordered field${\displaystyle \mathbb{F}}$ is infinitesimal if and only if itsabsolute value is smaller than any element of${\displaystyle \mathbb{F}}$ that is of the form${\displaystyle \frac{1}{n}}$, for${\displaystyle n}$ a standardnatural number. Ordered fields that have infinitesimal elements are also callednon-Archimedean. More generally, nonstandardanalysis is any form of mathematics that relies onnonstandard models and thetransfer principle. A field that satisfies the transfer principle for real numbers is called areal closed field, and nonstandardreal analysis uses these fields asnonstandard models of the real numbers.

Robinson's original approach was based on these nonstandard models of the field of real numbers. His classic foundational book on the subjectNonstandard Analysis was published in 1966 and is still in print.^[8] On page 88, Robinson writes:

The existence of nonstandard models of arithmetic was discovered byThoralf Skolem (1934). Skolem's method foreshadows theultrapower construction [...]

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 articleHyperreal number for a discussion of some of the relevant ideas.

In this section we outline one of the simplest approaches to defining a hyperreal field${\displaystyle {}^{\ast }\mathbb{R}}$. Let${\displaystyle \mathbb{R}}$ be the field of real numbers, and let${\displaystyle \mathbb{N}}$ be thesemiring of natural numbers. Denote by${\displaystyle {\mathbb{R}}^{\mathbb{N}}}$ the set of sequences of real numbers. A field${\displaystyle {}^{\ast }\mathbb{R}}$ is defined as a suitable quotient of${\displaystyle {\mathbb{R}}^{\mathbb{N}}}$, as follows. Take a nonprincipalultrafilter${\displaystyle F\subseteq P(\mathbb{N})}$. In particular,${\displaystyle F}$ contains theFréchet filter. Consider a pair of sequences

${\displaystyle u=(u_n),v=(v_n)\in {\mathbb{R}}^{\mathbb{N}}}$

We say that${\displaystyle u}$ and${\displaystyle v}$ are equivalent if they coincide on a set of indices that is a member of the ultrafilter, or in formulas:

${\displaystyle \{n\in \mathbb{N}:u_n=v_n\}\in F}$

The quotient of${\displaystyle {\mathbb{R}}^{\mathbb{N}}}$ by the resulting equivalence relation is a hyperreal field${\displaystyle {}^{\ast }\mathbb{R}}$, a situation summarized by the formula${\displaystyle {}^{\ast }\mathbb{R}={\mathbb{R}}^{\mathbb{N}}/F}$.

There are at least three reasons to consider nonstandard analysis: historical, pedagogical, and technical.

Much of the earliest development of the infinitesimal calculus byNewton and Leibniz was formulated using expressions such asinfinitesimal number andvanishing quantity. These formulations were widely criticized byGeorge Berkeley and others. The challenge of developing a consistent and satisfactory theory of analysis using infinitesimals was first met by Abraham Robinson.^[6]

In 1958 Curt Schmieden andDetlef Laugwitz published an article "Eine Erweiterung der Infinitesimalrechnung"^[9] ("An Extension of Infinitesimal Calculus") which proposed a construction of a ring containing infinitesimals. The ring was constructed from sequences of real numbers. Two sequences were considered equivalent if they differed only in a finite number of elements. Arithmetic operations were defined elementwise. However, the ring constructed in this way containszero divisors and thus cannot be a field.

H. Jerome Keisler,David Tall, and other educators maintain that the use of infinitesimals is more intuitive and more easily grasped by students than the"epsilon–delta" approach to analytic concepts.^[10] This approach can sometimes provide easier proofs of results than the corresponding epsilon–delta formulation of the proof. Much of the simplification comes from applying very easy rules of nonstandard arithmetic, as follows:

infinitesimal × finite = infinitesimal

infinitesimal + infinitesimal = infinitesimal

together with thetransfer principle (discussed further below).

Another pedagogical application of nonstandard analysis isEdward Nelson's treatment of the theory ofstochastic processes.^[11]

Some recent work has been done in analysis using concepts from nonstandard analysis, particularly in investigating limiting processes of statistics and mathematical physics.Sergio Albeverio et al.^[12] discuss some of these applications.

There are two, main, different approaches to nonstandard analysis: thesemantic ormodel-theoretic approach and the syntactic approach. Both of these approaches apply to other areas of mathematics beyond analysis, including number theory, algebra and topology.

Robinson's original formulation of nonstandard analysis falls into the category of thesemantic approach. As developed by him in his papers, it is based on studying models (in particularsaturated models) of atheory. Since Robinson's work first appeared, a simpler semantic approach (due to Elias Zakon) has been developed using purely set-theoretic objects calledsuperstructures. In this approacha model of a theory is replaced by an object called asuperstructureV(S) over a setS. Starting from a superstructureV(S) one constructs another object*V(S) using theultrapower construction together with a mappingV(S) → *V(S) that satisfies thetransfer principle. The map * relates formal properties ofV(S) and*V(S). Moreover, it is possible to consider a simpler form of saturation calledcountable saturation. This simplified approach is also more suitable for use by mathematicians who are not specialists in model theory or logic.

Thesyntactic approach requires much less logic and model theory to understand and use. This approach was developed in the mid-1970s by the mathematicianEdward Nelson. Nelson introduced an entirely axiomatic formulation of nonstandard analysis that he calledinternal set theory (IST).^[13] IST is an extension ofZermelo–Fraenkel set theory (ZF) in that alongside the basic binary membership relation ∈, it introduces a new unary predicatestandard, which can be applied to elements of the mathematical universe together with some axioms for reasoning with this new predicate.

Syntactic nonstandard analysis requires a great deal of care in applying the principle of set formation (formally known as theaxiom of comprehension), which mathematicians usually take for granted. As Nelson points out, a fallacy in reasoning in IST is that ofillegal set formation. For instance, there is no set in IST whose elements are precisely the standard integers (herestandard is understood in the sense of the new predicate). To avoid illegal set formation, one must only use predicates of ZFC to define subsets.^[13]

Another example of the syntactic approach is theVopěnka's alternative set theory,^[14] which tries to find set-theory axioms more compatible with the nonstandard analysis than the axioms of ZF.

Abraham Robinson's bookNon-standard Analysis was published in 1966. Some of the topics developed in the book were already present in his 1961 article by the same title (Robinson 1961).^[15] In addition to containing the first full treatment of nonstandard analysis, the book contains a detailed historical section where Robinson challenges some of the received opinions on thehistory of mathematics based on the pre–nonstandard analysis perception of infinitesimals as inconsistent entities. Thus, Robinson challenges the idea thatAugustin-Louis Cauchy's "sum theorem" inCours d'Analyse concerning the convergence of a series of continuous functions was incorrect, and proposes an infinitesimal-based interpretation of its hypothesis that results in a correct theorem.

Abraham Robinson and Allen Bernstein used nonstandard analysis to prove that every polynomially compactlinear operator on aHilbert space has aninvariant subspace.^[16]

Given an operatorT on Hilbert spaceH, consider the orbit of a pointv inH under the iterates ofT. Applying Gram–Schmidt one obtains an orthonormal basis(e_i) forH. Let(H_i) be the corresponding nested sequence of "coordinate" subspaces ofH. The matrixa_i,j expressingT with respect to(e_i) is almost upper triangular, in the sense that the coefficientsa_i+1,i are the only nonzero sub-diagonal coefficients. Bernstein and Robinson show that ifT is polynomially compact, then there is a hyperfinite indexw such that the matrix coefficienta_w+1,w is infinitesimal. Next, consider the subspaceH_w of*H. Ify inH_w has finite norm, thenT(y) is infinitely close toH_w.

Now letT_w be the operator${\displaystyle P_w\circ T}$ acting onH_w, whereP_w is the orthogonal projection toH_w. Denote byq the polynomial such thatq(T) is compact. The subspaceH_w is internal of hyperfinite dimension. By transferring upper triangularisation of operators of finite-dimensional complex vector space, there is an internal orthonormal Hilbert space basis(e_k) forH_w wherek runs from1 tow, such that each of the correspondingk-dimensional subspacesE_k isT-invariant. Denote byΠ_k the projection to the subspaceE_k. For a nonzero vectorx of finite norm inH, one can assume thatq(T)(x) is nonzero, or|q(T)(x)| > 1 to fix ideas. Sinceq(T) is a compact operator,(q(T_w))(x) is infinitely close toq(T)(x) and therefore one has also|q(T_w)(x)| > 1. Now letj be the greatest index such that. Then the space of all standard elements infinitely close toE_j is the desired invariant subspace.

Upon reading a preprint of the Bernstein and Robinson paper,Paul Halmos reinterpreted their proof using standard techniques.^[17] Both papers appeared back-to-back in the same issue of thePacific Journal of Mathematics. Some of the ideas used in Halmos' proof reappeared many years later in Halmos' own work on quasi-triangular operators.

Other results were received along the line of reinterpreting or reproving previously known results. Of particular interest is Teturo Kamae's proof^[18] of theindividual ergodic theorem or L. van den Dries andAlex Wilkie's treatment^[19] ofGromov's theorem on groups of polynomial growth. Nonstandard analysis was used by Larry Manevitz andShmuel Weinberger to prove a result in algebraic topology.^[20]

The real contributions of nonstandard analysis lie however in the concepts and theorems that utilize the new extended language of nonstandard set theory. Among the list of new applications in mathematics there are new approaches to probability,^[11]hydrodynamics,^[21] measure theory,^[22] nonsmooth and harmonic analysis,^[23] etc.

There are also applications of nonstandard analysis to the theory of stochastic processes, particularly constructions ofBrownian motion asrandom walks. Albeverio et al.^[12] have an introduction to this area of research.

In terms of axiomatics, Boffa’s superuniversality axiom has found application as a basis for axiomatic nonstandard analysis.^[24]

As an application tomathematical education,H. Jerome Keisler wroteElementary Calculus: An Infinitesimal Approach.^[10] Coveringnonstandard calculus, it develops differential and integral calculus using the hyperreal numbers, which include infinitesimal elements. These applications of nonstandard analysis depend on the existence of thestandard part of a finite hyperrealr. The standard part ofr, denotedst(r), is a standard real number infinitely close tor. One of the visualization devices Keisler uses is that of an imaginary infinite-magnification microscope to distinguish points infinitely close together.

This section should include a summary ofcriticism of nonstandard analysis. SeeWikipedia:Summary style for information on how to incorporate it into this article's main text.(June 2020)

Despite the elegance and appeal of some aspects of nonstandard analysis, criticisms have been voiced, as well, such as those byErrett Bishop,Alain Connes, andPaul Halmos.

Given any setS, thesuperstructure over a setS is the setV(S) defined by the conditions

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

${\displaystyle V_{n+1}(S)=V_n(S)\cup \mathrm{\wp }(V_n(S)),}$

${\displaystyle V(S)=\bigcup _{n\in \mathbf{N}}{V}_n(S).}$

Thus the superstructure overS is obtained by starting fromS and iterating the operation of adjoining thepower set ofS and taking the union of the resulting sequence. The superstructure over the real numbers includes a wealth of mathematical structures: For instance, it containsisomorphic copies of allseparablemetric spaces andmetrizable topological vector spaces. Virtually all of mathematics that interests an analyst goes on withinV(R).

The working view of nonstandard analysis is a set*R and a mapping* :V(R) →V(*R) that satisfies some additional properties. To formulate these principles we first state some definitions.

Aformula hasbounded quantification if and only if the only quantifiers that occur in the formula have range restricted over sets, that is are all of the form:

${\displaystyle \mathrm{\forall }x\in A,\mathrm{\Phi }(x,{\alpha }_1,\dots ,{\alpha }_n)}$

${\displaystyle \mathrm{\exists }x\in A,\mathrm{\Phi }(x,{\alpha }_1,\dots ,{\alpha }_n)}$

For example, the formula

${\displaystyle \mathrm{\forall }x\in A,\mathrm{\exists }y\in 2^B,x\in y}$

has bounded quantification, theuniversally quantified variablex ranges overA, theexistentially quantified variabley ranges over the powerset ofB. On the other hand,

${\displaystyle \mathrm{\forall }x\in A,\mathrm{\exists }y,x\in y}$

does not have bounded quantification because the quantification ofy is unrestricted.

A setx isinternal if and only ifx is an element of *A for some elementA ofV(R). *A itself is internal ifA belongs toV(R).

We now formulate the basic logical framework of nonstandard analysis:

• Extension principle: The mapping * is the identity onR.
• Transfer principle: For any formulaP(x₁, ...,x_n) with bounded quantification and with free variablesx₁, ...,x_n, and for any elementsA₁, ...,A_n ofV(R), the following equivalence holds:

${\displaystyle P(A_1,\dots ,A_n)⟺P(\ast A_1,\dots ,\ast A_n)}$

• Countable saturation: If {A_k}_{k ∈N} is a decreasing sequence of nonempty internal sets, withk ranging over the natural numbers, then

${\displaystyle \bigcap _k{A}_k\ne \mathrm{\varnothing }}$

One can show using ultraproducts that such a map * exists. Elements ofV(R) are calledstandard. Elements of*R are calledhyperreal numbers.

The symbol*N denotes the nonstandard natural numbers. By the extension principle, this is a superset ofN. The set*N −N is nonempty. To see this, apply countablesaturation to the sequence of internal sets

${\displaystyle A_n=\{k\in {}^{\ast }\mathbf{N}:k\ge n\}}$

The sequence{A_n}_{n ∈N} has a nonempty intersection, proving the result.

We begin with some definitions: Hyperrealsr,s areinfinitely closeif and only if

${\displaystyle r\cong s⟺\mathrm{\forall }\theta \in {\mathbf{R}}^{+},|r-s|\le \theta }$

A hyperrealr isinfinitesimal if and only if it is infinitely close to 0. For example, ifn is ahyperinteger, i.e. an element of*N −N, then1/n is an infinitesimal. A hyperrealr islimited (orfinite) if and only if its absolute value is dominated by (less than) a standard integer. The limited hyperreals form a subring of*R containing the reals. In this ring, the infinitesimal hyperreals are anideal.

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

Example: The plane(x,y) withx andy ranging over*R is internal, and is a model of plane Euclidean geometry. The plane withx andy restricted to limited values (analogous to theDehn plane) is external, and in this limited plane the parallel postulate is violated. For example, any line passing through the point(0, 1) on they-axis and having infinitesimal slope is parallel to thex-axis.

Theorem. For any limited hyperrealr there is a unique standard real denotedst(r) infinitely close tor. The mappingst is a ring homomorphism from the ring of limited hyperreals toR.

The mapping st is also external.

One way of thinking of thestandard part of a hyperreal, is in terms ofDedekind cuts; any limited hyperreals defines a cut by considering the pair of sets(L,U) whereL is the set of standard rationalsa less thans andU is the set of standard rationalsb greater thans. The real number corresponding to(L,U) can be seen to satisfy the condition of being the standard part ofs.

One intuitive characterization of continuity is as follows:

Theorem. A real-valued functionf on the interval[a,b] is continuous if and only if for every hyperrealx in the interval*[a,b], we have:*f(x) ≅ *f(st(x)).

Similarly,

Theorem. A real-valued functionf is differentiable at the real valuex if and only if for every infinitesimal hyperreal numberh, the value

${\displaystyle f^{\prime }(x)=\mathrm{st}\left(\frac{{}^{\ast }f(x+h)-{}^{\ast }f(x)}{h}\right)}$

exists and is independent ofh. In this casef′(x) is a real number and is the derivative off atx.

It is possible to "improve" the saturation by allowing collections of higher cardinality to be intersected. A model isκ-saturated if whenever${\displaystyle \{A_i{\}}_{i\in I}}$ is a collection of internal sets with thefinite intersection property and${\displaystyle |I|\le \kappa }$,

${\displaystyle \bigcap _{i\in I}{A}_i\ne \mathrm{\varnothing }}$

This is useful, for instance, in a topological spaceX, where we may want|2^X|-saturation to ensure the intersection of a standardneighborhood base is nonempty.^[25]

For any cardinalκ, aκ-saturated extension can be constructed.^[26]

• E. E. Rosinger, [math/0407178].Short introduction to Nonstandard Analysis. arxiv.org.

• Quotations related toNonstandard analysis at Wikiquote
• The Ghosts of Departed Quantitiesby Lindsay Keegan.Archived 25 April 2018 at theWayback Machine

• Nonstandard analysis
• Real closed field
• Infinity

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