uu.seUppsala University Publications
Change search
ReferencesLink to record
Permanent link

Direct link
Ultrapowers as sheaves on a category of ultrafilters
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2004 (English)In: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 43, no 7, 825-843 p.Article in journal (Refereed) Published
Abstract [en]

In the paper we investigate the topos of sheaves on a category of ultrafilters. The category is described with the help of the Rudin-Keisler ordering of ultrafilters. It is shown that the topos is Boolean and two-valued and that the axiom of choice does not hold in it. We prove that the internal logic in the topos does not coincide with that in any of the ultrapowers. We also show that internal set theory, an axiomatic nonstandard set theory, can be modeled in the topos.

Place, publisher, year, edition, pages
2004. Vol. 43, no 7, 825-843 p.
National Category
Algebra and Logic
URN: urn:nbn:se:uu:diva-91028DOI: 10.1007/s00153-004-0228-0OAI: oai:DiVA.org:uu-91028DiVA: diva2:163602
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

Open Access in DiVA

No full text

Other links

Publisher's full text
By organisation
Department of Mathematics
In the same journal
Archive for mathematical logic
Algebra and Logic

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Altmetric score

Total: 143 hits
ReferencesLink to record
Permanent link

Direct link