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Ultrasheaves and double negation
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2004 (English)In: Notre Dame Journal of Formal Logic, ISSN 0029-4527, Vol. 45, no 4, 235-245 p.Article in journal (Refereed) Published
Abstract [en]

Moerdijk has introduced a topos of sheaves on a category of filters. Following his suggestion, we prove that its double negation subtopos is the topos of sheaves on the subcategory of ultrafilters—the ultrasheaves. We then use this result to establish a double negation translation of results between the topos of ultrasheaves and the topos on filters.

Place, publisher, year, edition, pages
2004. Vol. 45, no 4, 235-245 p.
National Category
Algebra and Logic
URN: urn:nbn:se:uu:diva-91029DOI: 10.1305/ndjfl/1099238447OAI: oai:DiVA.org:uu-91029DiVA: diva2:163603
Available from: 2003-11-25 Created: 2003-11-25 Last updated: 2013-06-20Bibliographically approved
In thesis
1. Ultrasheaves
Open this publication in new window or tab >>Ultrasheaves
2003 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis treats ultrasheaves, sheaves on the category of ultrafilters.

In the classical theory of ultrapowers, you start with an ultrafilter and, given a structure, you construct the ultrapower of the structure over the ultrafilter. The fundamental result is Los's theorem for ultrapowers giving the connection between what formulas are satisfied in the ultrapower and in the original structure. In this thesis we instead start with the category of ultrafilters (denoted U). On this category U we build the topos of sheaves on U (the ultrasheaves), which we think of as generalized ultrapowers.

The theorem for ultrapowers corresponding to Los's theorem is Moerdijk's theorem, first proved by Moerdijk for the topos Sh(F) of sheaves on filters. In the thesis we prove that Los's theorem follows from Moerdijk's theorem. We also investigate the exact relation between the topos of ultrasheaves and Moerdijk's topos Sh(F) and prove that Sh(U) is the double negation subtopos of Sh(F).

The connection between ultrapowers and ultrasheaves is investigated in detail. We also prove some model theoretic results for ultrasheaves, for instance we prove that they are saturated models. The Rudin-Keisler ordering is a tool used in set theory to study ultrafilters. It has a strong relationship to the category U. Blass has given a model theoretic characterization of this ordering and in the thesis we give a new proof of his result.

One common use of ultrapowers is to give non-standard models. In the thesis we prove that you can model internal set theory (IST) in the ultrasheaves. IST, introduced by Nelson, is a non-standard set theory, an axiomatic approach to non-standard mathematics.

Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen, 2003. 54 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 30
Logic, symbolic and mathematical, 03G30, 03C20, 03H05, Matematisk logik
National Category
Algebra and Logic
urn:nbn:se:uu:diva-3762 (URN)91-506-1716-8 (ISBN)
Public defence
2003-12-18, Room 111, Building 1, Polacksbacken, Uppsala, 13:15
Available from: 2003-11-25 Created: 2003-11-25Bibliographically approved

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