"The language of infinitesimals is compatible with mathematical rigor" (Abraham Robinson).
"There is good reason to believe that nonstandard analysis, in one version or another, will be the analysis of the future."(Kurt Gödel)
Non-Standard Analysis
Introduction
The nonstandard analysis −also called "unconventional"−, devised by Abraham Robinson in the 1960s, is considered to be the culmination of a long history of the formalization, abstraction and conceptualization of numbers of infinite type.
The objectives of this theory are as follows:
To provide a precise meaning to something that was always confusing: everything related to concepts that included infinity: infinitely large, infinitely small, infinitely close, etc. numbers.
To provide a system of calculus with such numbers so that one could operate with them in the same way as with real numbers.
To achieve a more integrated, simpler, clearer and more intuitive mathematics, with shorter and more synthetic expressions than with standard analysis, and more practical.
Main concepts
Defines a new type of numbers, the hyperreal numbers (symbolized by *R), which generalize the conventional real numbers (R).
A hyperreal (or nonstandard) number is defined by an infinite (right-handed) sequence of real numbers. When all the components of the sequence are the same real number r, one has the equivalent of the real number r or "standard". If the components are not equal, the number is "non-standard". R is the set of standard numbers and *R−R is the set of non-standard numbers.
The operations and laws of traditional arithmetic also apply to hyperreals. In the operations of addition, subtraction, multiplication, and division of hyperreals, these operations are performed with each of the components of the sequence. The result is another hyperreal. The set of hyperreal numbers is an algebraic structure called a commutative ring.
A hyperreal number is infinitesimal if it is less than all conventional positive real numbers.
A hyperreal number r is finite if there exists a natural number n such that −n < r < n. A nonfinite hyperreal number is infinite.
If a hyperreal number r is infinitesimal, then its inverse 1/r is infinite and vice versa.
Every finite hyperreal number r has a single standard part std(r) and an infinitesimal one, which is r−std(r). If r is real, std(r) = r and its infinitesimal part is zero.
Two hyperreal numbers r and s are infinitely close, r≈s, if r−s is an infinitesimal.
The derivative of a function f(x) is defined as
df(x)/dx = std((f(x+dx)−f(x))/dx)
where dx is an infinitesimal and std is the function that extracts the standard part of the hyperreal.
Any property P(x) that is true for any standard number x, is also true for any nonstandard number (transfer principle).
If for every standard x there exists another standard y such that the statement P(x, y ) is true, then there exists a y (non-standard) such that P(x, y) is true as well (idealization principle).
For example,
P(x, y) is: "For all x>0 there exists an y intermediate between 0 and x (0<y<x)". This property, which is valid for every standard real number, is also valid for every nonstandard real number.
A hyperreal number r is bounded if there exist two real numbers a and b, such that a<r<b.
The "shadow" of a bounded hyperreal r, shadow(r), is the nearest real number, which is the standard part of r.
The "halo" of a hyperreal number r, halo(r), is the set of all hyperreals close to r. The set of all infinitesimals is halo(0).
The "galaxy" of a hyperreal number r, galaxy(r), is the set of all hyperreals that are at a bounded distance of r. The set of all bounded numbers is galaxy(0).
Nonstandard expressions are, in theory, simpler, more intuitive, and more practical than standard expressions. For example, the standard expression for continuity of a function f(x) at the point x0 is:
∀x ∀ε&>0 ∃δ>0 | |x − x0 |<δ → |f(x) − f(x0)|<ε
(for all x and for all ε>0, there exists δ>0 such that if |x − x0|<δ, then
|f(x) − f(x0)|<ε).
The non-standard expression is:
∀x (x ≈ x0 → f(x) ≈ f(x0))
(for all x infinitely close to x0, f(x) is infinitely close to f(x0)).
This expression is more intelligible because it uses the previous concept of infinitely close numbers.
The comparison relations (r<s, r≤s, etc. ) between two hyperreal numbers is not defined when all components of r and s satisfy that relation, but when it is satisfied for a specific set of indices called "ultrafilter". The order relation is total and there is a trichotomy law: for all r and s of *R, it is satisfied that r<s, r=s or r>s.
Allows to create superstructures (higher order structures).
The hyperfinite set is one of the most useful concepts of non-standard analysis, since it allows to study infinite structures satisfying various finiteness conditions.
Axiomatics
In first-order axiomatic systems, axioms can refer to properties of objects of the system, but cannot refer to higher-level objects (e.g., sets, sequences, subsets, sets of sets, etc.). In higher-order axiomatic systems, there may be axioms that refer to higher-level objects.
Examples:
∀x, y∈C (x+y = y+x)
(x+y = y+x for every pair of elements x, and of the set C)
This axiom is of first order, because it refers to the elements of the set C.
x<1/n ∀n∈N | n>1 → x=0
(if x<1/n for every natural number n>1, then x=0)
This axiom is of second order, because it refers to the elements of a subset of N (the numbers greater than 1).
The measure on a metric space E is an application between the subsets of E and R (the real numbers). It requires second-order objects, as well as their associated logic.
The axiomatic system itself established for the real numbers (R) is of second order.
But with nonstandard analysis one can work only with first-order axioms, thanks to the mechanisms of creating higher-order structures. For example, for the above example of the measure in a metric space, one creates the superstructure
S = P(E)×R
(Cartesian product of the power of E and the real numbers)
The measure in this metric space would be an element of S, and the axiomatic system would be of first order.
Limitations and problems of nonstandard analysis
Since its invention, nonstandard analysis has been attempted to be applied to a large number of domains: number theory, differential equations, probability theory, economic theory, mathematical physics, topology, functional analysis, dynamical systems, etc. It has been successfully used especially for the simplification of the proofs of some classical theorems and has even allowed new results to be found.
But the reality is that non-standard analysis has not been accepted as a practical foundation of Analysis, to the point that there is some skepticism about the survival of this theory, which has not become popular.
Among the many limitations and problems in this theory, we can point out:
It is not a global theory of mathematics. It is an extension of the traditional or standard Analysis that contemplates and integrates numbers of infinite type. The theory is generic, but not generic enough.
It is a somewhat contradictory theory. On the one hand it claims to achieve simplicity, but on the other hand it is complex, using too many concepts and impractical in general,
It does not include Cantor's transfinite numbers.
If hyperreal numbers exist, hyperreal numbers of higher order (hyper−hyperreal, etc.) should exist.
Many concepts of nonstandard analysis are informally defined.
To use the concept of sequence to define hyperreals is to distort the original concept of this mathematical structure. A sequence (finite or infinite) cannot conceptually be a number.
Traditional techniques of mathematical logic are applied, with their conceptual limitations.
An axiomatic approach is used, rather than a constructive approach, since it is recognized that sometimes formally constructing hyperrealities is complicated. What is done is to postulate by axioms their existence and properties.
The definite order relation between hyperreals is relative and not absolute, since it depends on a set of indices (the ultrafilter).
Infinite sequences of numbers can only be used descriptively in practice.
MENTAL vs. Non-Standard Analysis
MENTAL simplifies things so much that non-standard analysis becomes unnecessary. Everything is simpler and you don't need traditional axiomatics based on set theory, but use semantic axiomatics where you always use the same primitives (primary archetypes) to construct and describe expressions. It uses the just, primary and essential concepts from which other expressions and derived concepts, including real number structures, can be generated.
With MENTAL:
All kinds of expressions, numeric and symbolic, can be specified.
You can define infinitesimal and infinite expressions.
You can use predicates of any order.
Some examples of MENTAL relative to hyperreal
Definition of infinitesimal.
(ε*ε = 0)
Infinitely close numbers.
⟨( (r1 ≈ r2) ↔ abs(r1−r2 = ε) )⟩
(abs indicates absolute value)
Standard and non-standard parts.
If (r = r1 + r2*ε), the standard part of r is r1. And the non-standard part is r2*ε. The latter is also an infinitesimal, since (r2*ε)^2 = 0.
Continuity of a function at x0.
⟨( (x ≈ x0) → (f(x) ≈ f(x0) )⟩
This expression is even simpler than the one used by the non-standard analysis.
