Not standard analyzes

The birth of differential and infinitesimal calculus at the 17th century led to the introduction and the use of infinitely small quantities. Leibniz or Euler made great use of it. However, they could not clarify nature fully even these infinitely smalls. Their use disappeared at the 19th century with the development of the rigor in Analysis, since Cauchy until Weierstrass or Dedekind.

It was necessary to await second half of the 20th century so that a rigorous introduction of the infinitely smalls is proposed. In 1960, Abraham Robinson defines the infinitely smalls by a Ultra-power of $\backslash \; mathbb\; R$, thus giving birth to a new theory, the analysis nonstandard . A little later Nelson provides a presentation of the more accessible nonstandard analysis, based on the axiomatic of Zermelo-Frankel to which a new predicate is added: the standard predicate. The behavior of this new predicate is based on 3 new axioms:

1. the axiom of idealization
2. the axiom of standardization
3. the axiom of

transfer These 3 Axiome S are more known under the name STI.

The direction of the standard qualifier given by these axioms is that of object pertaining to the perceptible horizon, nonstandard as being beyond the perceptible horizon. A unit can thus be standard or not standard (one also says charmed), it cannot be both. The usual objects of traditional mathematics will be standard (1, 2, $\backslash \; pi$,…). The small infiniments or infiniments large introduced will be not standard.

Interest of the nonstandard analysis

There are two types of applications:

• It was established that a traditional statement, having a demonstration within the framework of the nonstandard analysis, was true within the framework of traditional mathematics. The situation is completely comparable with the mathematicians of before 1800, who were authorized to use the imaginary numbers provided that the end result is quite real. The nonstandard analysis thus makes it possible to give new demonstrations (often simpler) of traditional theorems.
• the nonstandard analysis makes it possible moreover to handle the new concepts of infinitely small number or infinitely great which posed so many problems to the mathematicians and who had been banished analysis. It is thus more general than the traditional analysis, just as the complex analysis is more general than the real analysis.
• However, the nonstandard analysis had to date little influence. Few new theorems were developed by means of this one, and for the moment, it constitutes primarily a rewriting of the whole of the analysis by means of new concepts.

Axioms

• One is placed within the framework of the set theory of Zermelo-Fraenkel (that which we all practice without the knowledge, like Mr. the Jordan).

• the objects or the units defined by this theory will be qualified interns or traditional. It is the case of all the objects and units usual which we know: $\backslash \; pi,\; E,\; 2,\; \backslash \; mathbb\; NR,\; \backslash \; mathbb\; R,\; \backslash \; mathbb\; C$…
• One introduces a new predicate, foreigner with the theory of Zermelo-Fraenkel, and which applies to the preceding units and internal objects. Such a unit or object could be qualified of standard or nonstandard. For example, one will be able to speak about standard entirety and nonstandard entirety. The “standard” word is not defined, not more than are not defined the words “together” or “membership”. They are what is called primitive concepts. One explains only the way in which one can use this new concept, by means of the axioms which follow.

Axiom of idealization

That is to say R (X, there) a “traditional” relation in a unit E. By traditional relation, one understands a relation not utilizing the new predicate “ standard ” in his statement. It is thus about a usual relation of our mathematics of the every day.

The axiom of idealization affirms that the two following proposals are equivalent:

1. For each standard part finished F, there exists an element X noted $x\_F$ such as R (X, there) for all pertaining one to F.

2. There exists the there an element X of E such as R (X, there) for all there standard .

The axiom means that, to find an element X which checks a property relative to all the elements there standard, it is necessary and it is enough to find such a X relating to the elements there of any finished standard unit.

Example 1: There exists an entirety higher than all the standard entireties

We want to show that: there exists X whole, such as, for all there standard entirety, X > Y. Is thus R (X, there) defined by: X is whole and is whole there and X > Y. Proposal 1 of the axiom of idealization is well checked: if F is finished (standard or not besides), there exists well an entirety X superior with the entireties there elements of F. Consequently, the axiom of idealization states that proposal 2 is also checked and this one corresponds to our statement.

There thus exists an entirety X superior with all the standard integers. This entirety will be thus not standard, if not, it would be higher than itself. We thus have just shown that there exists at least a nonstandard entirety. The entireties higher than X are a fortiori not standard, if not, X would be higher to them. For this reason, as a whole $\backslash \; mathbb\; N$ of the entireties, the nonstandard entireties are also qualified the inaccessible ones, or the unlimited ones, or infinitely greats. The “unlimited” term is perhaps badly selected. It could make believe that such entireties are infinite. But all the entireties are finished! We thus prefer the term of inaccessible or infinitely great. The nonstandard entireties are also called hypernaturels.

Example 2: Any infinite unit has a nonstandard element

Let us consider relation X different from there in a unit E infinite. For each finished part standard F, we find an element X noted $x\_F$ pertaining to E such as X is different from there for all there pertaining to F, since E is infinite.

The axiom of idealization then provides the existence of an element charmed (or not standard) X pertaining to E and different of all the elements standard there pertaining to E.

One from of deduced the following property:

In any infinite unit, there is at least a charmed element.

and by contraposition:

If all the elements of a unit are standard, this unit E is finished. (1)

Example 3: Theorem of Nelson

This theorem states that, if E is a unit, there exists a finished part X of E containing all the standard elements of E. the standard elements of a unit are thus of finished number. One defines for that the relation R (X, there) following: X is included in E, X is finished and is element of E so there, then is element of X there. Proposal 1 of the axiom of idealization is well vérifée for very finished part F (standard or not besides) by taking X the intersection of F and E. Consequently, proposal 2 of the axiom of idealization makes it possible to validate the theorem of Nelson.

It will be noted that part X data by the axiom is an internal or traditional part. It is not necessarily limited to the only standard elements of E, because, a priori, the collection of the standard elements, definite starting from the nontraditional relation “standard being” is an external object, i.e. foreign with usual mathematics. Indeed, the relation " to be standard" does not form part of the relations to which the axioms of ZFC apply, which wants to say that there do not exist overall containing only the standard entireties. Thus, in the entireties, a unit X container all the standard entireties is form {0, 1,2,…, N } with N not standard, and this unit contains also nonstandard entireties.

Axiom of transfer

As soon as all the parameters $E\_i$ of a traditional value F have standard values:

Pour any X standard, F (X, $E\_1$,…, $E\_n$) if and only so for any X, F (X, $E\_1$,…, $E\_n$)

In other words, to check that a usual formula depending on standard parameters is true for any X, it is enough to check it for any X standard. Intuitively, we can reach only the standard elements, and it is them which will enable us to check a traditional formula. This axiom can be also expressed (by negation):

Il exists X standard, F (X, $E\_1$,…, $E\_n$) if and only if there exists X, F (X, $E\_1$,…, $E\_n$)

If a traditional property is true for X, then it is true for X standard. Here are some consequences. Most important is the fact that if a mathematical object is defined in a traditional way in a single way starting from standard objects, it is necessarily standard. It is thus the case of $\backslash \; varnothing,\; 0,1,2,\; \backslash \; pi,\; E,\; I,\; 2,\; \backslash \; mathbb\; NR,\; \backslash \; mathbb\; R^n$ for N standard . In the same way, if E and F are standard units, it is the same of their intersection, their meeting, their product, the whole of the applications of E in F, of the whole of the parts of E. If has and B is two standard numbers, it is the same of ab , has + B , has - B , has / B , etc If N is standard, he is the same of N +1 or I _N = {1,…, N }. If has is a limited standard part of $\backslash \; mathbb\; R$, Sup has and Inf has are standard. If F is a standard function (i.e. definite on standard units and of standard graph), then the image of a standard element is standard.

Lastly, this axiom makes it possible to show that, to see that two standard units are equal, it is enough to check that they have the same standard elements. Thus, the only standard part of $\backslash \; mathbb\; N$ containing all the standard entireties is $\backslash \; mathbb\; N$ itself. On the other hand, there exist nonstandard parts containing all the standard entireties, namely the parts {0, 1,2,…, N } with N not standard.

Axiom of standardization

Either E a standard unit, or P an unspecified property, utilizing or not the postulate “ standard ”. Then:

Il exists a unit has standard such as for any X standard, X belongs to has if and only if X belongs to E and checks P (X)

This axiom is of interest only if the property P is nontraditional (it uses the postulate “ standard ”). Has is not other than a standard unit whose standard elements are the standard elements of E checking the property P. It may be that has of other elements, but they will be not standard. In addition, a standard unit being defined in a single way by its standard elements, it results from it that has is single. It is called standardized collection { X element of E | P ( X )} who, a priori, is not a whole with direction ZFC. Intuitive interpretation that one can give to this axiom is the following: the collection { X element of E | P ( X )} is not directly accessible for us. We can conceive only its standardized. We insist on the fact that, if the property P uses the postulate “ standard ”, this property is foreign with axiomatic of Zermelo-Fraenkel (since the word “ standard ” does not form part of this axiomatic), and thus that the collection { X element of E | P ( X )} is not a whole within the meaning of Zermelo-Fraenkel, this is why let us qualify we it collection. (more technically, the property P is not necessarily collectivizing, and the notation { X element of E | P ( X )} is formally as illegal as would be it, for example, { X | X = X } to indicate the whole of all the units).

For example, let us consider E = $\backslash \; mathbb\; N$, and P ( X ) the property X is standard. The collection { X element of E | P ( X )} is the collection of the standard elements. Its standardized is a standard unit containing all the standard elements of $\backslash \; mathbb\; N$. We already saw that it was about $\backslash \; mathbb\; N$ itself.

Let us consider E now = $\backslash \; mathbb\; N$, and P ( X ) the property X is not standard. The collection { X element of E | P ( X )} is the collection of the nonstandard elements. Its standardized is the empty set.

The numbers analyzes not standard of it

Entireties

Let us recall that we qualify intern or traditional properties or units not using the word “ standard ”. We call external or not traditional the properties using this mot. All the traditional known properties remain valid Analyzes not standard of it. Thus, $\backslash \; mathbb\; N$ checks the axiom of recurrence, provided that this axiom is applied to a traditional property.

On the other hand, the predicate standard being nontraditional, the axiom of recurrence does not apply to it. Thus, 0 are standard; if N is standard, N + 1 too. However, there exist entireties nonstandard superiors with all the standard entireties. Such nonstandard entireties are called infinitely great.

Entire standard is lower than entire not standard. If N is not standard, it of in the same way elements higher than N and of N - 1. One can see $\backslash \; mathbb\; N$ as follows:

0 1 2 3……… N -1 N N +1…

whole standard followed by the entireties nonstandard

One cannot speak about the smallest nonstandard entirety, not more than we cannot speak about the greatest standard entirety, because these units are not traditional, and one cannot thus apply to them the traditional properties of $\backslash \; mathbb\; N$.

If P is an unspecified property, it is shown that $\backslash \; mathbb\; N$ checks the principle of recurrence restricted according to:

if P (0) is true, and so for all N standard , P ( N ) implies P ( N +1), then for all N standard , P ( N ) is vérifié.

Realities

It is shown that one can partitionner the unit $\backslash \; mathbb\; R$ of realities in:

• the infinitesimal ones or infinitely smalls, inferiors in absolute value with all real standard. With share 0, they are not standard. X - infinitesimal is noted there X ≈ there . It is said that X and is infinitely close there.
• the unlimited ones, superior in value absolute with any reality (or entire) standard. They are not standard. Their opposite are infinitesimal.
• the appreciable ones. The appreciable ones and the infinitesimal ones belong to limited realities.

For example: 0,000… 01 is infinitely small if the number of 0 is an entirety infinitely great. This number is then infinitely close to 0.

If N is an entirety infinitely great, then 1 N is infinitely small.

It is also shown that, for each reality limited X , there exists a single reality ° X standard such as the difference X - ° X is infinitesimal. ° X is called standard part of X .

For example, 0,3333 ..... 333 where the number of 3 is an entirety infinitely great is a reality limited not standard, of which the standard part is 1/3.

Any limited reality breaks up in a single way in the standard form + infinitesimal.

Realities infinitely close to a given reality constitute the halation of this reality.

The continuations analyzes not standard of it

We will give nontraditional properties of the continuations, which, in the case of the standard continuations, will coincide with usual properties.

Convergence of a continuation

For a standard continuation $(a\_n)$, there is equivalence between:

1. the continuation $(a\_n)$ converges towards L
2. L is standard and, for all N infinitely great, $a\_n$ ≈ L

Indeed, if $(a\_n)$ is standard and converges towards L , its limit is standard (by transfer) and checks:

pour all ε > 0, there exists NR such as, for all N > NR, | $a\_n$ - L | < ε

By transfer, one has then:

pour all standard ε > 0, there exists NR standard such as, for all N > NR, | $a\_n$ - L |

If one takes N infinitely great, N is then higher than NR thus | $a\_n$ - L | < ε, and this inequality being checked for all standard ε, one has well $a\_n$ ≈ L

Reciprocally, if, for all N infinitely great, $a\_n$ ≈ L with L standard , then:

pour all standard ε > 0, there exists NR such as, for all N > NR, | $a\_n$ - L | < ε

It is indeed enough to take NR infinitely great.

and by transfer:

pour all ε > 0, there exists NR such as, for all N > NR, | $a\_n$ - L | < ε

what is the definition of convergence.

It will be noted well that stated equivalence is valid only for the standard continuations. If one defines $a\_n\; indeed\; =\; (-\; 1)\; ^n\; \backslash \; alpha$ with α infinitely small, then $a\_n$ ≈ 0 for all N and yet the continuation $(a\_n)$ does not converge (but this continuation is not a standard continuation).

Convergence of a under-continuation

For a continuation $(a\_n)\; standard$, there is equivalence between:

1. there exists a under-continuation of $(a\_n)$ which converges towards L
2. L is standard and there exists unlimited N such as $(a\_n)$ ≈ L

Indeed, if L is limiting of a under-continuation of $(a\_n)$, then L is standard by transfer, and for all ε > 0, there exists an infinity of N such as | $a\_n$ - L | < ε. This property is thus true for ε infinitely small, and as it is checked by an infinity of N and that there exists only one finished number of standard entireties, there thus exists N infinitely great such as | $a\_n$ - L | < ε. But as ε is infinitely small, that means that $(a\_n)$ ≈ L .

Reciprocally, if there exists unlimited N such as $(a\_n)$ ≈ L , then:

for all standard ε > 0, for any NR standard, there exists N > NR, | $a\_n$ - L | < ε

er by transfer:

for all ε > 0, any NR, there exists N > NR, | $a\_n$ - L | < ε

what expresses that L is Valeur of adherence of the continuation $(a\_n)$ and in this case, there exists well a under-continuation of $(a\_n)$ which converges.

One from of deduced the theorem from Bolzano-Weierstrass , which expresses, that, from any limited real continuation, one can extract a under-continuation which converges. By transfer, it is enough to show this theorem on the standard continuations. That is to say thus $(a\_n)$ a limited standard continuation. All its terms are limited because, by transfer, one can take one raising and one undervaluing of $(a\_n)\; standard$. One takes then unlimited N and L = standard °$a\_n$ left $a\_n$. One applies equivalence shown then previously, property 2 being checked.

Continuation of Cauchy

For a continuation $(a\_n)\; standard$, there is equivalence between:

1. $(a\_n)$ is a Suite of Cauchy
2. for all N and p unlimited, $a\_n$ ≈ $a\_p$

The demonstration follows a step comparable with those of the preceding paragraphs.

Let us show that, in $\backslash \; mathbb\; R$, any continuation of Cauchy converges. By transfer, it is enough to show this property on the standard continuations. That is to say $(a\_n)$ such a continuation. It is limited: indeed, there is only one finished number of standard entireties, and all the $a\_n$ with unlimited N are in the same halation of the one of them. By transfer, the terminal can be selected standard. All the terms of the continuation are thus limited. One takes L then = standard °$a\_p$ left $a\_p$ with p unlimited. Then, for all unlimited N , $a\_n$ ≈ $a\_p$ ≈ L , therefore the continuation converges towards L .

The functions analyzes not standard of it

Continuity

The continuity of a function in $\backslash \; mathbb\; R$ is defined more simply with the nonstandard analysis. For a standard function, there is equivalence between

1. $f$ is continuous
2. for all $y$ infinitely small and for all standard $x$, $f\; (x+y)$ is infinitely close to $f\; (X)$.

One shows the theorem of the intermediate values in the following way. That is to say F continuous on a segment with F ( has ) < 0 and F ( B ) > 0. Then there exists C between has and B such as F ( C ) = 0. Indeed, by transfer, it is enough to show this theorem for F , has and B standard . Either NR an unlimited entirety and $x\_k\; =\; has\; +\; K\; \{Ba\; \backslash \; over\; NR\}$ for K between 0 and NR. If K is the first K for which $f\; (x\_k)\; \backslash \; Ge\; 0$ then one will take for C the standard part of $x\_K$. There is effect C infinitely near to $x\_K$ and $x\_\; \{K-1\}$, so that F ( C ) will be infinitely close to positive or null reality $f\; (x\_K)$ and infinitely near to negative reality $f\; (x\_\; \{K-1\})$. Being standard, F ( C ) is null.

One shows in a comparable way that F admits a maximum and a minimum.

Uniform continuity

For a standard function, there is equivalence between

1. $f$ is uniformly continuous
2. for all $y$ infinitely small and for all $x$, $f\; (x+y)$ is infinitely close to $f\; (X)$.

For example, the function which with X associates X ² is continuous, since, if X is standard and there infinitely small, one a:

( X + there ) ² = X ² + 2 xy + there ² ≈ X ² since X being limited, xy is infinitely small, like there ²

On the other hand, this function is not uniformly continuous since, if X is infinitely great and if there = 1 X , then ( X + there ) ² = X ² + 2 + there ² which is not infinitely close to X ².

On a segment, any function continues F is uniformly continuous. By transfer, it is enough to show this property for F , has and B standard . The elements of the segment all are then limited, therefore admit a whole a standard part. If X is element of, ° X its part standard and there infinitely small, one a:

F ( X + there ) = F (° X + Z ) with Z = there + X - ° X infinitely small

thus F ( X + there ) ≈ F (° X ) ≈ F ( X ) by continuity of F in ° X

Derivation

For a standard function definite on a standard interval of $\backslash \; mathbb\; R$, and for X standard ₀ there is equivalence between:

1. F is derivable in X ₀ of derived L
2. for any X infinitely close to X ₀, $$ ≈ L , with L standard .

Integration

For a standard function F on, '' B '' = I standard, there is equivalence between

1. F integrable within the meaning of Riemann
2. for any subdivision of has = X ₀ < X ₁ <… < x_n = B with x_i ≈ X _{I +1}, there exist two functions in staircase φ and ψ relating to the subdivision so that, for all X of I, φ ( X ) ≤ F ( X ) ≤ ψ ( X ), and $\backslash \; int\_a^b\; \backslash \; psi\; -\; \backslash \; phi$ ≈ 0. One poses $\backslash \; int\_a^b\; f$ then the standard part of $\backslash \; int\_a^b\; \backslash \; phi$ or $\backslash \; int\_a^b\; \backslash \; psi$.

Various concepts

We give below examples of equivalent in nonstandard analysis of notions of the traditional analysis, when they are applied to standard objects. The purpose of those are to show the extent of the fields to be explored.

• simple Convergence towards F of a succession ( F _N ) of functions: for all N infinitely great and all X standard , F _N ( X ) ≈ F ( X ).

• uniform Convergence towards F of a succession ( F _N ) of functions: for all N infinitely great and all X , F _N ( X ) ≈ F ( X ).
• Compactness of a space K: any point of K is almost stantard (a point is almost standard if it is infinitely close to a standard point)
• Complétude of a space E: any quasi standard point is almost standard (a point X is quasi standard so for any R standard, X is at a distance lower than R of a standard point)

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