Hyperreal number

The hyperreal numbers are an extension of the set of real numbers that allow, among other things, to formalize some operations with infinitesimals, and prove some classic results of real analysis in a simpler way.

The system of hyperreal numbers is a way to treat infinite and infinitesimal quantities. Non-standard hyperreals, ${displaystyle} {^{mathbb {R} }$, are an extension of the real numbers ${displaystyle mathbb {R} }$ containing larger numbers

${displaystyle 1+1+cdots +1,}$ (for any finite number of terms).

Such a number is infinite, and its reciprocal is infinitesimal. The term "hyper-real" it was introduced by Edwin Hewitt in 1948.

As an algebraic structure, they are a non-Archimedean and metrically incomplete field that contains the Archimedean and complete set identifiable with the real numbers. Formally, they can be constructed in a totally rigorous way from a first-order axiomatization of the real numbers. Said axiomatization is a non-categorical theory and therefore admits several non-isomorphic models, one of them the standard real numbers and another of them identifiable with the hyperreal ones. In addition, if it is intended to avoid the theory of models, the theory of real numbers can be extended by means of an abstract predicate (semantically interpretable as "x is a standard real number") and three additional axioms that describe said predicate (these predicates allow us to characterize the difference between a standard real number and an unconventional hyperreal one).

History

The concept of hyperreal number comes from Non-standard analysis, a domain that was developed in the 1970s by Abraham Robinson, although it has a background in the work of Wilhelmus Luxemburg in the 1960s and Edwin Hewitt (1948). The non-standard analysis aims, and achieves, to rigorously justify the use of infinite and infinitesimal numbers. The set of reals plus these new elements are called hyperreal numbers and are designated ${displaystyle} {^{mathbb {R} }$, fulfilling that ${displaystyle mathbb {R} subset !{}^{*}mathbb {R}}}}$. Somehow, the ancient Greek mathematicians used an intuitive approach to hyperreal numbers, although in a totally intuitive and non-rigrious way. For these mathematicians, a length a was infinitesimal compared to b if multiplying it for any integer would never get over b: 2a3a4a...1000a...n·a... they're all inferior to b (with n any integer). This definition is the very denial of the fundamental property that says that the set of real numbers is archemedian.

Between the Renaissance and the 18th century, infinitesimals were used again and Gottfried Leibniz proposed a theory, built from an infinite number “greater than all existing integers”. This theory did not have solid logical foundations, but it allowed the calculations that physicists needed, especially in differential equations. Nonstandard analysis formalizes Leibniz's notions of arithmetic of infinitesimals and infinities as nonstandard hyperreal numbers. In addition to infinitesimals and unlimited (infinite), the limited (complement of the previous set) and appreciable (neither infinitesimal nor unlimited) are defined. From these four sets we have the following Leibniz rules for the arithmetic operations of these sets:

+/-	infinitesimal	limited	appreciable	unlimited
infinitesimal	infinitesimal	limited	appreciable	unlimited
limited	limited	limited	limited	unlimited
appreciable	appreciable	limited	limited	unlimited
unlimited	unlimited	unlimited	unlimited	?

For multiplication, the Leibniz rules are as follows:

x	infinitesimal	limited	appreciable	unlimited
infinitesimal	infinitesimal	infinitesimal	infinitesimal	?
limited	infinitesimal	limited	limited	?
appreciable	infinitesimal	limited	appreciable	unlimited
unlimited	?	?	unlimited	unlimited

These heuristic rules continued to be used well into the 18th century, when the theory of limits was invented and perfected, rendering them useless. Cauchy, Dedekind, Cantor, Weierstrass, Bolzano and Heine, among other mathematicians, had taken care to specify in a completely rigorous way the concepts of continuity and limit. These mathematicians developed a rigorous formalism that made it possible to eliminate numerous aporias and paradoxes from analysis (see for example 1 − 2 + 3 − 4 + · · ·). The price of this rigor was a heavy and unintuitive formalism, although more productive and free of contradictions. In the 19th and 20th centuries, dreams were made of inventing mathematics that would make room for the longed-for infinite numbers (big or small).

The temptation was always to add these ill-defined quantities to the set of real numbers, but the problem was that one then had to find out if the theorems valid for the real numbers were valid or not for the hyperreal numbers. Naturally, it was never achieved, because it was not the proper method.

Construction

There are three conceivable ways to arrive at constructing a set like the hyperreal numbers:

• La Direct construction adding a number of hyperreal numbers to the real and postulating rules ad hoc specific to arithmetic operations (historically this is the way in which the infinitese were introduced). The infinitese would be smaller numbers than any conventional real number, and their respective inverses would correspond to "infinite" or "not matched") numbers.
• Like extension of the theory of the real, this approach that possibly allows to handle hyperreal numbers more easily and demonstrate results consists of introducing a new preaching ${displaystyle mathrm {est} (cdot)}$ and three new axioms called "principle of transfer", "principle of idealization"and "principle of standardization".
• Like non-standard model of the theory of real numbers.

Intuitive rendering

The direct construction model is the least formal of the construction procedures, and therefore formally the weakest. However, many of the insights that led to the other formal constructions came from generalizing the possible properties of infinitesimals. The intuitive representation in this section illustrates the properties of systems that can be formally constructed by other methods and that could be reached by generalizing direct addition.

The following figure represents the hyperreal line on three different scales: ω is any infinite number (such as those that can be shown to exist in a non-standard model of the theory of the real ones) and ε is an infinitesimal, also any. Both are positive.
To go from one line to the next we enlarge the scale by a factor of infinity. In the first line, the finite numbers cannot be distinguished because they are all infinitely close to zero, as if glued together. In the second it is the infinitesimals that cannot be glimpsed, and the infinities are logically an infinite distance from zero.

The infinites of this theory have nothing to do with those invented by Georg Cantor, in the context of the ordinales and the cardinals. (see infinite numbers). In fact, Cantor, who invented (in the West) the notion of infinite number was only interested in the integers, while the non-standard analysis deals with the real ones. If ω designates Cantor's first infinity, then ${displaystyle omega over 2}$ and

${displaystyle {sqrt {7}}omega }$

simply have no meaning in your theory.

Direct construction

Hyperreal numbers can be conceived as an infinite and stratified set of copies of a set of limited hyperreal numbers ${displaystyle mathbb {L} =mathrm {Lim} ({}^{}mathbb {R}}}$. Note that this set contains all ordinary real numbers ${displaystyle mathbb {R} subset mathbb {L} }$ in addition to their respective "halos" (see below). The halo or monad of a real number x is a set of infinitely close hyperreal numbers x:

${displaystyle {text{monad}}(x)={yin mathbb {R} ^{}mid x-y{text{ is infinitesimal}}}}}}}$

The notion of infinitesimal can be rigorously defined in the language of extended real number theory with the predicate "standard" (See later). In fact, all the infinitesimal numbers turn out to be all the non-zero hyperreal numbers that make up the monad of the real number 0:

${displaystyle epsilon mathrm {infinitesimal} Leftrightarrow epsilon in {text{monad}}(0)}$

The set of real numbers together with their monads satisfies the relation:

${displaystyle mathbb {R} subset bigcup _{xin mathbb {R}}{text{monad}}(x)subseq mathbb {L} }$

For any infinitesimal number ${displaystyle epsilon neq 0}$ the unlimited hyperreal number ${displaystyle h_{epsilon }=1/epsilon notin mathbb {R} }$and a "transmitted copy" ${displaystyle mathbb {L} }$:

${displaystyle mathbb {L} _{h_{epsilon }}:=left{xin} {^{{*}mathbb {R} ⋅exists yin mathbb {R}: x=y+{frac {1}{epsilon }{right}}{right$

Finally, the set of hyperreals can be conceived as the set of all the copies transferred with the previous one:

${displaystyle}{mathbb {R} =left(bigcup _{epsilon} mathbb {L} _{h_{epsilon }}}right)bigcup mathbb {R} }$

Extension from the theory of reals

Another logical possibility offered by mathematical logic. Conventional real numbers are a possible realization of the so-called first-order theory of real numbers. This theory consists of a set of axioms expressible in a first-order formal language. The real numbers commonly used in solving problems of mathematical analysis satisfy said axioms, as well as all the theorems logically deducible from said theorems by means of the deduction rules of said formal language. If the axioms are slightly modified or some new symbols are introduced into the basic alphabet of the original formal language, a model can be obtained that includes numbers with the properties traditionally attributable to infinitesimal numbers.

To build the system of hyperreal numbers according to this approach, it is not necessary to touch the construction of the sets of numbers, but rather the logical-formal language that serves as the foundation for that construction (that is, the axioms that the model sought must satisfy). This can be done from an axiomatic formalization of the theory of numerical sets like the one that can be obtained from the Zermelo-Fraenkel axioms. From this theory, the compactness theorem of first order logic can be used to obtain a model with the desired properties. This model also allowed adding new axioms to the old axioms to the theory consistent with the previous ones. Specifically, A. Robinson invented a new unary predicate: "standard" and from there two cases arise: a number x is standard or it is not. In this connection, the following distinction between internal and external property is very important:

A property or proposition is internal if it can be expressed in the theory of Zermelo-Fraenkel, that is to say if it does not require the word Standard or one of its derivatives to be defined. The word is also used Standard to qualify a formula internal which can cause confusion: a formula is Standard Yeah. No. contains the word Standard...

A formula is external when you cannot write without using the word Standard or one of its derivatives.

Three conditions were then imposed on this predicate (called transfer, idealization, and standardization) to ensure the existence of new numbers, not standards, with the appropriate properties, worthy of infinitesimals and infinities, more specifically the transfer property was formulated.

Axioms of transfer and idealization

This transfer property is as follows:

If for any x standard, P (x) it is true (P is an internal proposition) then P (x) is true for any x (whether standard or non-standard):

${displaystyle forall ^{st}P (forall ^{st}x P(x))Longrightarrow (forall x P(x))}$

This property means that all classical rules, which are true in usual mathematics, generalize without any change to non-standard objects. In other words, you don't have to prove them again. For example, let P (x) be the proposition: if x > 0 then there exists y such that 0 < and < x. We know that P (x) is always true in the usual reals (for y it is enough to take x/2). P is also an internal proposition. Consequently, P is also valid for all non-standard reals. Transference is often used in its opposite form:

${displaystyle forall ^{st}Q (exists x Q(x))Longrightarrow (exists ^{st}x Q(x))}}$

Which can be paraphrased like this: if there is an element that checks for an internal property, then there is a standard element that also checks for it. The idealization property is as follows (with P an inner statement):

If for everything x Standard exists and such as P (x, y) be true, then there is a and for everything x standard, P (x, y) be true:

${displaystyle forall ^{st}P (forall ^{st}x exists y P(x,y)Longrightarrow (exists y forall ^{st}x P(x,y))}{$

It's been permuted. x and andand the new and That's it. ideal in the sense that works with all x. For example, let's take the P previous: P (x, y) means: 0 and. x. We know that for any xStandard, there is a and between him and 0, therefore there must be a and ideal always between 0 and any x  0 standard. In other words, there is a different number of zero but less than any positive real. This number is by definition an infinitesimal, and its nature is described as follows: ${displaystyle andapprox 0}$

In the same way it is shown that there are infinite numbers (which have nothing to do with infinite ordinales or infinite cardinals): For all x Standard exists and (e.g. x + 1), then there is a and ideal greater than all x standards: is by definition an infinite number, which is denoted ${displaystyle andapprox +infty}$. The property of standardization is technical, and of little interest at the moment.

Hyperreals as a non-standard model

The set of hyperreals constitutes a model in the sense of the model theory of the axioms of the first-order theory that axiomatically defines the real numbers. This theory is not logically complete, so it admits various non-isomorphic models. Ordinary real numbers are a model for such theory, another possible model is hyperreal numbers, which satisfy the axioms of the axiomatic theory of real numbers but some valid properties in the standard model are not valid in the non-standard model (although both models satisfy all the deducible theorems of the axiomatic theory).

A natural way to build the non-standard model of hyperreal numbers from the standard model (ordinary real numbers) is to define a first-order language ${displaystyle {mathcal {L}}_{mathbb {R} }}$ where besides signs for quantifiers, relationships/preached and functions are included an infinite number of constants c_a (one for every real number built in theory). That language can formalize the ordinary real numbers that constitute a possible model of such language. Now consider the set of sentences expressed in that language given by:

${displaystyle sigma:=mathrm {Th} mathbb {R} cup {c_{}{a}{mathbb {R}subset mathrm {Sent} ({mathcal {L}}}_{mathbb {R}}}}}}$

Note that this set is infinite, since there is an infinite number of constants and where the parts that define it are:

${displaystyle mathrm {Th} mathbb {R} ={pin mathrm {Sent} ({mathcal {L}}}_{mathbb {R})}{mathbb {R}} mathbb {R} vDash p}}}}}$

${displaystyle x_{1},}$ is any variable of language ${displaystyle {mathcal {L}}_{mathbb {R} }}$.

, is the set of sentences expressed in ${displaystyle {mathcal {L}}_{mathbb {R} }}$ which are valid in the model of ordinary real numbers. Given any finite subset of the previous ${displaystyle Sigma _{0}subset Sigma }$ is satisfyable only to assign to the variable x₁ value b high enough inside ${displaystyle mathbb {R} }$ (the standard model) given the finitude of the subset is always possible to satisfy this condition:

${displaystyle mathbb {R} vDash Sigma _{0}quad [[b]]}$

In addition, the compactness theorem guarantees the existence of a model that contains the previous one where it is satisfied that:

${displaystyle}{^{*}mathbb {R} vDash Sigma quad [[H]]$

It can be verified that this model contains unbounded elements such as H, and therefore, this model can be interpreted as the set of hyperreal numbers in which the same theorems that satisfied the ordinary real numbers are satisfied.

Applications

Continuity and uniform continuity

To see the benefit that can be derived from non-standard analysis, let's compare the expression for continuity at point x:

(1)${displaystyle forall epsilon /20050 exists alpha /20050 forall y (left️y-xright ingredientalpha Longrightarrow leftѕf(y)-f(x)right inherentepsilon)quad {mbox{(classical expression}}}}}$

(2)${displaystyle forall y ( yapprox xLongrightarrow f(y)approx f(x)quad {mbox{(expression in non-standard analysis)}}}}$

The non-standard formula is much more intuitive and practical. In general, hyperreal numbers make it possible to suppress many quantifiers, that is, to lower the complexity of the formulas.

Proof of equivalence:

The classic expression is shape ${displaystyle forall epsilon \ P(epsilon)}$with P a standard proposition (as long as f be a standard function as well). So by transfer equals ${displaystyle forall ^{st}epsilon P(epsilon)}$.

P(ε) is shape ${displaystyle exists }$α Q(α, ε). By transfer, too, equals ${displaystyle exists ^{st}}$α Q(α, ε).

So far we have obtained the equivalence between (1) and:

(1')${displaystyle forall ^{st}epsilon /20050 exists ^{st}alpha  y (left/25070/y-xright knowing \alpha Longrightarrow leftωf(y)-f(x)right knowingepsilon)}$

Now, by definition any infinitesimal is less than α and ε that are strictly positive standards. Then yes x ${displaystyle approx }$ and then wholesome ${displaystyle approx }$ 0 then UDY - x UD α.

By implication of (1') is obtained Δf (y) - f (x)UD ε. As this is true for any standard ε δ0, then Δf (y) - f (x)Δ is infinitesimal, which means that f (y)${displaystyle approx }$f(x) We just proved that (1') implies (2(c):

The reciprocal is very similar: Suppose (2), and choose ε 한 0 standard. Then any infinitesimal α agrees on (1(c):

If UDY - xUD α ${displaystyle approx }$ 0 then and ${displaystyle approx }$ x then (2): f (y) ${displaystyle approx }$ f (x) then Șf (y) - f (x) ${displaystyle approx }$ 0 and therefore Δf (y) - f (x)Δ ε.

By transfer there is also a standard α that suits, which gives (1').

Continuity in everything ${displaystyle scriptstyle mathbb {R} }$ equals (by transfer) to continuity in all its standards:

(3)${displaystyle forall ^{st}x forall y ( yapprox xLongrightarrow f(y)approx f(x)$

The uniform continuity over the interval I = R is expressed as follows:

${displaystyle (4)quad quad forall epsilon /20050 forall x exists alpha /20050 forall y (left/25070/y-xright knowingalpha Longrightarrow leftficientf(y)-f(x)right knowingepsilon}$

expression in non-standard analysis:

${displaystyle (5)quad quad forall x forall y ( yapprox xLongrightarrow f(y)approx f(x)$

The only difference between (3) and (5) is that in uniform continuity x It doesn't have to be standard. They are not equivalent because you cannot apply the transfer here: the ${displaystyle approx }$ make it not a standard formula.

Consider the function f: ${displaystyle {begin{matrix}mathbb {R} &longrightarrow &mathbb {R}{xlongmapsto &x^{2}end{matrix}}}}}}}$

To show your continuity, let's take x standard, and therefore finite, e and = x + ε con ${displaystyle epsilon approx 0}$ infinitesimal. (laughs) ${displaystyle andapprox x}$).

Then ${displaystyle f(y)=y^{2}=(x+epsilon)^{2}=x^{2} +2xepsilon +epsilon ^{2}=x^{2}+epsilon approx x^{2}=f(x)}}$ because 2x + ε is a finite number that, multiplied by an infinitesimal, ε gives an infinitesimal. This shows continuity.

But f is not uniformly continuous: if we take this time an infinite x: x = ω and ${displaystyle epsilon ={frac {1}{omega }}}}$ infinitesimal, then:

${displaystyle f(omega +epsilon)=omega ^{2}+2omega cdot {frac {1}{omega }{omega ^{1}{omega ^{2}}}{omega ^⁄2}}{2⁄2app+{frac {1}{1}{omega ^{x x x}{2}{2}}{$. There is no simpler proof.

Limits

The limit of a succession corresponds to an infinite range value of this. More precisely, ${displaystyle (u_{n})_{nin mathbb {n}} }$ a convergent succession (standard) towards l eventually infinite. So, for everything. ${displaystyle omega approx +infty u_{omega }approx} l$

The notions of continuity and limits are formally very similar, in fact a limit can be interpreted as a continuity at an infinite point. So the tests are essentially the same.

The classic expression of ${displaystyle lim _{nto infty }u_{n}=l}$ It's, uh, stop. l finite:

${displaystyle forall epsilon 0 exists Nin mathbb {Nforall n(n/2005N)Longrightarrow (IVAu_{n}-lepsilon)}}}}$

Properties

• The whole ${displaystyle} {^{mathbb {R} }$ is an ordained body not arquimedian, and as a consequence it is not complete (all ordered and complete body is arquimedian).
• The set of limited hyperreal numbers ${displaystyle mathrm {Lim} ({}{mathbb {R}}}}$ It is a subanillo (this set includes ordinary reals and infinitesimals, the inverses of infinitesimals are unlimited elements and therefore do not belong to this subanillo).
• The whole ${displaystyle mathrm {inf}$ formed by all infinitesimal elements forms a maximal ideal of the previous ring ${displaystyle mathrm {Lim} ({}{mathbb {R}}}}$
• The quotient ring ${displaystyle A:=mathrm {Lim} ({}^{*}mathbb {R}}/mathrm {inf} ({}^{*}mathbb {R}}}}}}$ it is indeed an orderly and archery body. In fact it can be shown that such quotient ring can be identified with actual numbers ${displaystyle Aapprox mathbb {R} }$.
• The whole ${displaystyle mathbb {n} subsetneq {}^{mathbb {n}subset} {^{mathbb {R} }$ of natural hyperreals has a cardinal at least ${displaystyle 2^{aleph _{0}}}}$, while the set of hyperreal rationals has a cardinal at least ${displaystyle 2^{2^{aleph}}}}$. This proves that hyperreal numbers are much more numerous than real numbers.

Generalization

Hyperreal numbers can be extended to algebra- or field-structured number systems by various types of constructions. For example, the superreal numbers are an extension of the hyperreal numbers, and the surreal numbers in turn extend the superreal numbers. The existence of a chain of inclusions can be shown as follows:

${displaystyle mathbb {R} mathrm {(reales)} subset {^{*}mathbb {R} mathrm {(hyperreales)} subset mathrm {superreales} subset mathrm {surreales} }$