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A logical presentation of the continuous functionals

Published online by Cambridge University Press:  12 March 2014

Erik Palmgren  and
Viggo Stoltenberg-Hansen
Erik Palmgren
Affiliation:
Department of Mathematics, Uppsala University, Po Box 480, S-751 06 Uppsala, Sweden, E-mail: palmgren@math.uu.se
Viggo Stoltenberg-Hansen
Affiliation:
Department of Mathematics, Uppsala University, Po Box 480, S-751 06 Uppsala, Sweden, E-mail: viggo@math.uu.se
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Extract

The Kleene-Kreisel continuous functionals [6, 7] have been given several alternative characterisations: using Kuratowski's limit spaces (Scarpellini [20]), using their generalisation, filter spaces (Hyland [4]), via hyperfinite functionals (Normann [11]) and perhaps most elegantly using Scott-Ershov domains (Ershov [3], later generalised by Berger [1]). We propose to add yet another characterisation to this list, which may be called model-theoretic in contrast to the others, but which is in fact closely related to Ershov's approach.

We use the notion of logically presented domains developed in Palmgren and Stoltenberg-Hansen [17] (originating in the work of [15]). Certain logical types, i.e., finitely consistent sets of formulas, over the full type structure built from ℕ correspond to the continuous functionals in the sense of Kreisel, or to Kleene's associates, while the elements realising the types correspond to functionals having associates (cf. [6]). To define these—the total types—we make a nonstandard extension of the full type structure. The set of nonstandard elements is sufficiently rich to single out the total types. The nonstandard extension also makes it possible to relate the logical presentation to Ershov's approach. The logical form of the domain constructions allows us to use a (generalised) Fréchet power as a nonstandard extension. This extension is constructive, in the sense that it avoids the axiom of choice, as distinguished from the one employed in [11].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 1021 - 1034
Copyright
Copyright © Association for Symbolic Logic 1997

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References

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