Fine Art


In abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by H. Garth Dales and W. Hugh Woodin as a generalization of the hyperreal numbers and primarily of interest in non-standard analysis, model theory, and the study of Banach algebras. The field of superreals is itself a subfield of the surreal numbers.

Dales and Woodin's superreals are distinct from the super-real numbers of David O. Tall, which are lexicographically ordered fractions of formal power series over the reals.[1]
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 \( \Bbb{R}\), so that F is not order isomorphic to \( \Bbb{R} \).

If the prime ideal P is a maximal ideal, then F is a field of hyperreal numbers (Robinson's hyperreals being a very special case).


David Tall, "Looking at graphs through infinitesimal microscopes, windows and telescopes," Mathematical Gazette, 64 22– 49, reprint at


Dales, H. Garth; Woodin, W. Hugh (1996), Super-real fields, London Mathematical Society Monographs. New Series 14, The Clarendon Press Oxford University Press, ISBN 978-0-19-853991-9, MR 1420859

L. Gillman and M. Jerison: Rings of Continuous Functions, Van Nostrand, 1960.

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

Retrieved from ""
All text is available under the terms of the GNU Free Documentation License

Home - Hellenica World