Hyperreal number
In mathematics, the hyperreal numbers, denoted, are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number is said to be finite when for some integer. Similarly, is said to be infinitesimal when for all positive integers. The term "hyper-real" was introduced by Edwin Hewitt in 1948.
The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about are also valid in. For example, the commutative law of addition,, holds for the hyperreals just as it does for the reals; since is a real closed field, so is. Similarly, since for all integers, one also has for all hyperintegers. The transfer principle for ultrapowers is a consequence of Łoś's theorem of 1955.
Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson outlined.
The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis. One immediate application is the definition of the basic concepts of analysis such as the derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of becomes
for an infinitesimal, where denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable infinite sum.
Transfer principle
The idea of the hyperreal system is to extend the real numbers to form a system that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra. Any statement of the form "for any number " that is true for the reals must also be true for the hyperreals. For example, the axiom that states "for any number, " still applies. The same is true for quantification over several numbers, e.g., "for any numbers and,." 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 " may not carry over. The only properties that differ between the reals and the hyperreals are those that rely on quantification over sets, or other higher-level structures such as functions and relations, which are typically constructed out of sets. Each real set, function, and relation has its natural hyperreal extension, satisfying the same first-order properties. The kinds of logical sentences that obey this restriction on quantification are referred to as statements in first-order logic.The transfer principle, however, does not mean that and have identical behavior. For instance, in there exists an element ω such that
but there is no such number in. This is possible because the nonexistence of cannot be expressed as a first-order statement.
Use in analysis
Informal notations for non-real quantities have historically appeared in calculus in two contexts: as infinitesimals, like, and as the symbol, used, for example, in limits of integration of improper integrals.As an example of the transfer principle, the statement that for any nonzero number,, is true for the real numbers, and it is in the form required by the transfer principle, so it is also true for the hyperreal numbers. This shows that it is not possible to use a generic symbol such as for all the infinite quantities in the hyperreal system; infinite quantities differ in magnitude from other infinite quantities, and infinitesimals from other infinitesimals.
Similarly, the casual use of is invalid, since the transfer principle applies to the statement that zero has no multiplicative inverse. The rigorous counterpart of such a calculation would be that if is a non-zero infinitesimal, then is infinite.
For any finite hyperreal number, the standard part,, is defined as the unique closest real number to ; it necessarily differs from only infinitesimally. The standard part function can also be defined for infinite hyperreal numbers as follows: If is a positive infinite hyperreal number, set to be the extended real number, and likewise, if is a negative infinite hyperreal number, set to be .
Differentiation
One of the key uses of the hyperreal number system is to give a precise meaning to the differential operator as used by Leibniz to define the derivative and the integral.For any real-valued function the differential is defined as a map which sends every ordered pair to an infinitesimal
Note that the very notation "" used to denote any infinitesimal is consistent with the above definition of the operator for if one interprets to be the function then for every the differential will equal the infinitesimal.
A real-valued function is said to be differentiable at a point if the quotient
is the same for all nonzero infinitesimals If so, this quotient is called the derivative of at.
For example, to find the derivative of the function, let be a non-zero infinitesimal. Then,
The use of the standard part in the definition of the derivative is a rigorous alternative to the traditional practice of neglecting the square of an infinitesimal quantity. Dual numbers are a number system based on this idea. After the third line of the differentiation above, the typical method from Newton through the 19th century would have been simply to discard the term. In the hyperreal system,
, since is nonzero, and the transfer principle can be applied to the statement that the square of any nonzero number is nonzero. However, the quantity is infinitesimally small compared to ; that is, the hyperreal system contains a hierarchy of infinitesimal quantities.
Using hyperreal numbers for differentiation allows for a more algebraically manipulable approach to derivatives. In standard differentiation, partial differentials and higher-order differentials are not independently manipulable through algebraic techniques. However, using the hyperreals, a system can be established for doing so, though resulting in a slightly different notation.
Integration
Another key use of the hyperreal number system is to give a precise meaning to the integral sign ∫ used by Leibniz to define the definite integral.For any infinitesimal function, one may define the integral as a map sending any ordered triple to the value
where is any hyperinteger satisfying
A real-valued function is then said to be integrable over a closed intervalif for any nonzero infinitesimalthe integral
is independent of the choice of If so, this integral is called the definite integral of on
This shows that using hyperreal numbers, Leibniz's notation for the definite integral can actually be interpreted as a meaningful algebraic expression.
Properties
The hyperreals form an ordered field containing the reals as a subfield. Unlike the reals, the hyperreals do not form a standard metric space, but by virtue of their order they carry an order topology.The use of the definite article the in the phrase the hyperreal numbers is somewhat misleading in that there is not a unique ordered field that is referred to in most treatments. However, a 2003 paper by Vladimir Kanovei and Saharon Shelah shows that there is a definable, countably saturated elementary extension of the reals, which therefore has a good claim to the title of the hyperreal numbers. Furthermore, the field obtained by the ultrapower construction from the space of all real sequences, is unique up to isomorphism if one assumes the continuum hypothesis.
The condition of being a hyperreal field is a stronger one than that of being a real closed field strictly containing. It is also stronger than that of being a superreal field in the sense of Dales and Woodin.
Development
The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert the existence of at least one infinitesimal number, and the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed.From Leibniz to Robinson
When Newton and Leibniz introduced differentials, they used infinitesimals and these were still regarded as useful by later mathematicians such as Euler and Cauchy. Nonetheless these concepts were from the beginning seen as suspect, notably by George Berkeley. Berkeley's criticism centered on a perceived shift in hypothesis in the definition of the derivative in terms of infinitesimals, where dx is assumed to be nonzero at the beginning of the calculation, and to vanish at its conclusion. When in the 1800s calculus was put on a firm footing through the development of the -definition of limit by Bolzano, Cauchy, Weierstrass, and others, infinitesimals were largely abandoned, though research in non-Archimedean fields continued.However, in the 1960s Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. Robinson developed his theory nonconstructively, using model theory; however it is possible to proceed using only algebra and topology, and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers per se, aside from their use in nonstandard analysis, have no necessary relationship to model theory or first order logic, although they were discovered by the application of model theoretic techniques from logic. Hyper-real fields were in fact originally introduced by Hewitt by purely algebraic techniques, using an ultrapower construction.