# Superreal number

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Superreal number

The superreal numbers are an extension of the real numbers, similar to the surreal numbers or hyperreal numbers, but comprising a more inclusive category than either one.

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 quotient field F of A is a superreal field if F strictly contains the real numbers $Bbb\left\{R\right\}$, so that F is not order isomorphic to $Bbb\left\{R\right\}$, 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.

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