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Order in open intervals of computable reals

Published online by Cambridge University Press:  01 February 1999

JAN VON PLATO
JAN VON PLATO
Affiliation:
University of Helsinki, Unioninkatu 40 B, 00014 University of Helsinki, Finland. E-mail: vonplato@helsinki.fi
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Abstract

In constructive theories, an apartness relation is often taken as basic and its negation used as equality. An apartness relation should be continuous in its arguments, as in the case of computable reals. A similar approach can be taken to order relations. We shall here study the partial order on open intervals of computable reals. Since order on reals is undecidable, there is no simple uniformly applicable lattice meet operation that would always produce non-negative intervals as values. We show how to solve this problem by a suitable definition of apartness for intervals. We also prove the strong extensionality of the lattice operations, where by strong extensionality of an operation f on elements a, b we mean that apartness of values implies apartness in some of the arguments: f(a, b)≠f(c, d) ⊃a≠c∨b≠d.

Most approaches to computable reals start from a concrete definition. We shall instead represent them by an abstract axiomatically introduced order structure.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 9 , Issue 1 , February 1999 , pp. 103 - 108
Copyright
1999 Cambridge University Press

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