At the beginning of calculus in the 17th century, people noticed the basic problems of this discipline.1. NewtonBoth Leibniz and Leibniz have used infinitesimal, especially Leibniz and his followers developed calculus theory on the basis of first order and higher order infinitesimal;They are fully allowed to introduceInfinitesimalandInfinity, and they are regarded as ideal elements similar to imaginary numbers, which obey the laws of ordinary real numbers.The symbols they used were widely used on the European continent.The superiority of these marks promoted the rapid development of calculus theory on the European continent at that time.Therefore, Robinson regarded Leibniz as the true pioneer of non-standard analysis.However, there is a significant internal contradiction in this theory - sometimes infinitesimal is regarded as non-zero and divisor, and sometimes it is regarded as zero and discarded.Limited to the conditions at that time, this contradiction could not be completely solved for a while, and it was inevitably criticized and attacked.BritishSubjective idealismPhilosopher BBerkeley(1685-1753) In 1734, the bishop wrote an article attacking infinitesimal as "the ghost of lost quantity".Until the 19th century, A. - L. Cauchy, B. Polzano and K（T.W.）WeierstrassThe limit theory has established a logically rigorous foundation for mathematical analysis, thus promoting the great development of mathematical analysis.
Since then, infinitesimal and infinity have no place in analysis any more, only such statements as "a variable tends to infinity" remain.Although the limit theory makes the mathematical analysis obtain logical preciseness, it loses the conciseness and intuitiveness of the infinitesimal method.Because the infinitesimal method is convenient to shorten the argument, "it is more suitable for the art of inventors".Many physicists, economists and engineers are still accustomed to using the infinitesimal method.However, mathematicians believe that there is no infinitesimal as a number in mathematical analysis.Until the 1960s, Robinson used mathematical logic to rigorously demonstrate the existence of infinitesimal, successfully solved Leibniz's problem of "infinitesimal contradiction", and createdNon-standard analysis。Then W. Luxembourg constructedNon-standard modelLater, we constructed a multi saturation model.Since then, non-standard analysis has developed rapidly and has been successfully applied to many aspects, such as point sets, topology, measure theoryfunction space 、probability theory、differential equation 、Algebraic number theory, fluid mechanicsquantum mechanics、Theoretical physicsAnd mathematical economy.Non standard analysis provides a good model for the commercial market with numerous small trade.Moreover, it is very effective to simulate the thermodynamic process of a gas under pressure in a container with an infinite boundary.Non standard analysis has made useful contributions to some difficult problems in some disciplines.For example, non-standard analysis methods are used to solve the problems that have not been solved for decadesHilbert spaceOf polynomial compact operators onInvariant subspaceExisting problems;For another example, Chinese mathematicians have given an effective method to solve the multiplication problem of generalized functions with non-standard analysis methods;For another example, French mathematiciansordinary differential equation A lot of significant results have been made on the singular perturbation of.It must also be pointed out that, except for standard analysis, there are still various attempts to make infinitesimal and infinity applicable in analysis.In this regard, the most effective ones are D. Ragwitz's infinite decimals and the research on generalized numbers proposed by Chinese scholars[1]。
definition
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American mathematical logician A. Robinson was founded in 1960.Robinson proved that real number structure
Structure that can be expanded to include infinite decimals and infinite numbers
, in a sense
And
They have the same nature.call
The number in is a hyperreal number. Figuratively speaking, an infinite decimal (whose absolute value is less than any real number) and an infinite number (whose absolute value is greater than any real number) are added to an ordinary real number.When two hyperreal numbers
And
When the difference is infinite, it is called
Infinitely close to
, recorded as
This is an equivalence relationship.Each equivalence class about this equivalence relation contains a unique standard real number
。call
Equivalent class
It's a list. The list is not
Is equal to
The super real number can perform four operations, which meet the usual operation rules, and can also have the order of size.Therefore, many concepts and theorems in standard analysis can be naturally extended to non-standard analysis.Such as interval
Expand to
，
The function in is extended to
, function
At standard point
Continuous can be defined as
When,
；function
stay
Upper uniformly continuous can be defined as when
，
，
Hour
[2]。
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Hyperreal numberThe system is designed to strictly deal with infinity（InfinityandInfinitesimal quantity）And put forward.sinceCalculusMathematicians, scientists and engineers (includingNewtonandLeibnizThe concept of infinitesimal has been widely used.A set of hyperreal numbers, orNon-standard real numberSet, marked as
, is one of the real number setsexpand；It contains a number that is greater than all numbers in the following forms:
This can be interpreted asInfinity；And their reciprocal is taken asInfinitesimal quantity。ℝ satisfies the following properties:First order propositionIf it is established, it is also established for ℝ.This property is called communication principle.For example, theAdditive commutative law
It is a first-order proposition about ℝ.Therefore, the following proposition is also true:
Is the concept of infinitesimal quantity strict?This problem can be traced back to ancient Greek mathematics: mathematicians such asEuclid、ArchimedesIn order to avoid the argument of infinitesimal quantity in some proofs to ensure the strictnessExhaustion methodAnd other explanation methods.Abraham Robinson proved in the 1960s that,
Hyperreal number systems are compatible if and only if real number systems are compatible[3]。
In other words, if you have no doubt about the use of real numbers, you can also use super real numbers with confidence.When dealing with mathematical analysis problems, the use of super real numbers, especially the transmission principle, is generally calledNon-standard analysis。
