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A generalized cut characterization of the fullness axiom in CZF
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2013 (English)In: Logic journal of the IGPL (Print), ISSN 1367-0751, E-ISSN 1368-9894, Vol. 21, no 1, 63-76 p.Article in journal (Refereed) Published
Abstract [en]

In the present note, we study a generalization of Dedekind cuts in the context of constructive Zermelo-Fraenkel set theory CZF. For this purpose, we single out an equivalent of CZF's axiom of fullness and show that it is sufficient to derive that the Dedekind cuts in this generalized sense form a set. We also discuss the instance of this equivalent of fullness that is tantamount to the assertion that the class of Dedekind cuts in the rational numbers, in the customary constructive sense including locatedness, is a set. This is to be compared with the situation for the partial reals, a generalization in a different direction: if one drops locatedness from the notion of a Dedekind real, then it becomes inherently impredicative. We make this precise, and further present the curiosity that the irrational Dedekind reals form a set already with exponentiation.

Place, publisher, year, edition, pages
2013. Vol. 21, no 1, 63-76 p.
Keyword [en]
Dedekind reals, axiom of choice, axiom of power set, constructive, predicative
National Category
Natural Sciences
URN: urn:nbn:se:uu:diva-194740DOI: 10.1093/jigpal/jzs022ISI: 000313837700006OAI: oai:DiVA.org:uu-194740DiVA: diva2:606504
Available from: 2013-02-19 Created: 2013-02-19 Last updated: 2013-02-19Bibliographically approved

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