Hyperreal number

In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. This principle allows true first order statements about R to be reinterpreted as true first order statements about *R.

Related Topics:
Mathematics - Non-standard analysis - Mathematical logic - Ordered field - Extension - Real number - Transfer principle - First order

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An important property of *R is that it has infinitely large as well as infinitesimal numbers, where an infinitely large number is one that is larger than all numbers representable in the form

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:1 + 1 + cdots + 1.

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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.

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However, a 2003 paper by Kanovei and Shelah shows that there is a definable, countably saturated (meaning ω-saturated, but not of course countable) elementary extension of the reals, which therefore has a good claim to the title of the hyperreal numbers.

Related Topics:
Shelah - Saturated - Elementary extension

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The condition of being a hyperreal field is a stronger one than that of being a real closed field strictly containing R. It is also stronger than that of being a superreal field in the sense of Dales and Woodin.

Related Topics:
Real closed field - Superreal field - Woodin

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The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis; some find it more intuitive than standard real analysis. When Newton and (more explicitly) 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 Berkeley, and when in the 1800s calculus was put on a firm footing through the development of the epsilon-delta definition of a limit by Cauchy, Weierstrass and others, they were largely abandoned.

Related Topics:
Analysis - Nonstandard analysis - Real analysis - Newton - Leibniz - Euler - Cauchy - Berkeley - 1800s - Calculus - Limit - Weierstrass

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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 the use of them in nonstandard analysis, have no necessary relationship to model theory or first order logic.

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