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Infinitesimal
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Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the "infinite-th" member of a series) have been used to express the idea of objects so small that there is no way to see them or to measure them. For everyday life, an infinitesimal object is an object which is smaller than any possible measure. When used as an adjective in the vernacular, "infinitesimal" means extremely small, but not necessarily "infinitely small".
Before the nineteenth century none of the mathematical concepts relating to infinitesimals as we know them today were formally defined, but many of these concepts were already there.
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Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the "infinite-th" member of a series) have been used to express the idea of objects so small that there is no way to see them or to measure them. For everyday life, an infinitesimal object is an object which is smaller than any possible measure. When used as an adjective in the vernacular, "infinitesimal" means extremely small, but not necessarily "infinitely small".
Before the nineteenth century none of the mathematical concepts relating to infinitesimals as we know them today were formally defined, but many of these concepts were already there. The founders of calculus, Leibniz, Newton, Euler, Lagrange, the Bernoullis and many others, used infinitesimals and achieved correct results even though no formal definition was available (similarly, there was no formal definition of real numbers at the time).
History of the infinitesimal
The notion of infinitesimally small quantities was discussed discussed by the Eleatic School. Archimedes, in The Method of Mechanical Theorems, was the first to propose a logically rigorous definition of infinite and infinitesimal numbers. His Archimedean property defines a number x as infinite if it satisfies the conditions |x|>1, |x|>1+1, |x|>1+1+1, ..., and infinitesimal if x?0 and the same set of conditions holds for 1/x. A number system is said to be Archimedean if it contains no infinite or infinitesimal members. In the ancient Greek system of mathematics, 1 represents the length of some line segment which has arbitrarily been picked as the unit of measurement. Thus the Archimedean definition really defines a notion of one number's being infinite or infinitesimal relative to another.
When Newton and Leibniz developed calculus, they made use of infinitesimals. The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work The Analyst. Although mathematicians, scientists, and engineers continued to use infinitesimals to produce correct results, it was not until the second half of the nineteenth century that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a limit. In the 20th century, it was found that infinitesimals could also be treated rigorously. Neither formulation is wrong, and both give the same results if used correctly.
Commonly used nonarchimedean systems
The hyperreal system
The most widespread technique for handling infinitesimals was developed by Abraham Robinson in the middle of the 20th century. The real numbers are extended to include infinitesimals, forming the hyperreals, without changing any of the elementary axioms of algebra. The only properties that differ between the reals and the hyperreals are those that rely on quantification over sets. Any statement of the form "for any number x..." that is true for the reals is also true for the hyperreals. For example, the axiom that states "for any number x, x+0=x" still applies. The same is true for quantification over several numbers, e.g., "for any numbers x and y, xy=yx." This ability to carry over statements from the reals to the hyperreals is called the transfer principle. However, statements of the form "for any set of numbers S ..." may not carry over.
The hyperreal system was specifically designed to be a conservative extension of the real number system in which it would be convenient to do analysis. The article Hyperreal number gives examples of the use of hyperreal numbers in calculus.
Surreal numbers
Another nonarchimedean system is Conway's surreal numbers. Compared to the hyperreals, the surreal numbers are more closely linked to the ordinal numbers, and the constructive approach can be applied to them more easily. However, they are less suitable for use in analysis because the transfer principle does not apply to them, and also because the existence of any particular surreal number, even one that has a direct counterpart in the reals, is not known a priori, and must be proved.
Smooth infinitesimal infinitesimals
Alternatively, we can have synthetic differential geometry or smooth infinitesimal analysis with its roots in category theory. This approach departs dramatically from the classical logic used in conventional mathematics by denying the law of excluded middle--i.e., not (a ? b) does not have to mean a = b. A nilsquare or nilpotent infinitesimal can then be defined. This is a number x where x2 = 0 is true, but x = 0 need not be true at the same time. With an infinitesimal such as this, algebraic proofs using infinitesimals are quite rigorous, including the one given above.
Logical properties
Proving or disproving the existence of infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used. We consider here systems where infinitesimals can be shown to exist.
In 1936 Maltsev proved the compactness theorem. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them. A consequence of this theorem is that if there is a number system in which it is true that for any positive integer n there is a positive number x such that 0 < x < 1/n, then there exists an extension of that number system in which it is true that there exists a positive number x such that for any positive integer n we have 0 < x < 1/n. The possibility to switch "for any" and "there exists" is crucial. The first statement is true in the real numbers as given in ZFC set theory : for any positive integer n it is possible to find a real number between 1/n and zero, but this real number will depend on n. Here, one chooses n first, then one finds the corresponding x. In the second expression, the statement says that there is an x (at least one), chosen first, which is between 0 and 1/n for any n. In this case x is infinitesimal. This is not true in the real numbers (R) given by ZFC. Nonetheless, the theorem proves that there is a model (a number system) in which this will be true. The question is: what is this model? What are its properties? Is there only one such model?
There are in fact many ways to construct such a one-dimensional linearly ordered set of numbers, but fundamentally, there are two different approaches:
- 1) Extend the number system so that it contains more numbers than the real numbers.
- 2) Extend the axioms (or extend the language) so that the distinction between the infinitesimals and non-infinitesimals can be made in the real numbers.
In 1960, Abraham Robinson provided an answer following the first approach. The extended set is called the hyperreals and contains numbers less in absolute value than any positive real number. The method may be considered relatively complex but it does prove that infinitesimals exist in the universe of ZFC set theory. The real numbers are called standard numbers and the new non-real hyperreals are called nonstandard.
In 1977 Edward Nelson provided an answer following the second approach. The extended axioms are IST, which stands either for Internal Set Theory or for the initials of the three extra axioms: Idealization, Standardization, Transfer. In this system we consider that the language is extended in such a way that we can express facts about infinitesimals. The real numbers are either standard or nonstandard. An infinitesimal is a nonstandard real number which is less, in absolute value, than any positive standard real number.
In 2006 developed an extension of Nelson's approach in which the real numbers are stratified in (infinitely) many levels i.e, in the coarsest level there are no infinitesimals nor unlimited numbers. Infinitesimals are in a finer level and there are also infinitesimals with respect to this new level and so on.
All of these approaches are mathematically rigorous.
See also
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