Superreal number
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The superreal numbers are a class of extensions of the real numbers, generalizing the hyperreal numbers.
[edit] Formal Definition
Suppose X is a Tychonoff space, also called a T3.5 space, and C(X) is the algebra of continuous real-valued functions on X. Suppose P is a prime ideal in C(X). Then the factor algebra A = C(X)/P is by definition an integral domain which is a real algebra and which can be seen to be totally ordered. The field of fractions F of A is a superreal field if F strictly contains the real numbers 
, so that F is not order isomorphic to .
If the prime ideal P is a maximal ideal, then F is a field of hyperreal numbers.
The terminology is due to Dales and Woodin.
Dales and Woodin's supperreals are different from David Tall's super-reals, which are lexicographically ordered fractions of formal power series over the reals[1].
[edit] References
- ^ David Tall, "Looking at graphs through infinitesimal microscopes, windows and telescopes," Mathematical Gazette, 64 22– 49, reprint at http://www.warwick.ac.uk/staff/David.Tall/downloads.html
- H. Garth Dales and W. Hugh Woodin: Super-Real Fields, Clarendon Press, 1996.
- L. Gillman and M. Jerison: Rings of Continuous Functions, Van Nostrand, 1960.
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