Superreal number

The superreal numbers comprise a more inclusive category than hyperreal number.

Suppose X is a Tychonoff space, also called a T_3.5 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 quotient field F of A is a superreal field if F strictly contains the real numbers ${\mathbb {R}}$, so that F is not order isomorphic to ${\mathbb {R}}$, though they may be isomorphic as fields.

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.

References

• 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.