Ultrapowers as Sheaves on a Category of Ultrafilters
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Ultrapowers as Sheaves on a Category of Ultrafilters
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| Id. |
41894945 |
| Idioma |
inglés
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| Titulo |
Ultrapowers as Sheaves on a Category of Ultrafilters |
| Autor(es) |
Filosofie Licentiatavhandling,Jonas Eliasson
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| Localización |
http://citeseer.ist.psu.edu/459440.html
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| Versión |
1.0 |
| Estado |
Final
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| Descripción |
In 1993 I. Moerdijk presented a new model of nonstandard arithmetic in the topos of sheaves on a category of filters, Sh($mathbb{F}$). This was later extended by E. Palmgren to a model of nonstandard analysis. The model in particular makes use of the sheaves ${}^*S$, which at any filter $mathcal{F}$ is the reduced power of the set $S$ over $mathcal{F}$, ${}^*S(mathcal{F})$. The details of this will be given in section 1.3. Before this, in section 1.1, we will give a short background to the subject of sheaves and logic and, in section 1.2, some preliminaries. In this paper we focus our attention on the sheaves on the subcategory of ultrafilters, Sh($mathbb{U}$). The category $mathbb{U}$ will be discussed in section 2. The sheaves of the form ${}^*S$ now, at an ultrafilter $mathcal{U}$, represents the ultrapower of $S$ over $mathcal{U}$, ${}^*S(mathcal{U})$. More details on the sheaves over $mathbb{U}$ can be found in section 3. In section 4 we study the internal logic in the topos of sheaves, which is classic since Sh($mathbb{U}$) is an atomic topos. We prove that this logic does not coincide with the logic in any of the ultrapowers ${}^*S(mathcal{U})$. The category of ultrafilters has a strong connection with ultrafilters under the Rudin-Keisler ordering, for instance we have $mathcal{U} leq mathcal{V}$ if and only if $ extup{Hom}_{mathbb{U}}(mathcal{V}, mathcal{U}) ot = emptyset$. In the paper we define the Rudin-Keisler ordering on Sh($mathbb{U}$) and study the consequences of it in our setting. In the paper we investigate the properties of Sh($mathbb{U}$). We establish two transfer principles: external transfer, which is corresponding to {L}o{'s} theorem, and an internal transfer principle. We show that the topos theoretic axiom of choice does not hold in Sh($mathbb{U}$) but establish some weak form of it and also prove some other properties similar to results proved by Palmgren about Sh($mathbb{F}$). In section 5 we show that the topos can be used to model Nelson's internal set theory (IST). IST is an axiomatic approach to nonstandard analysis, which adds to ZFC a undefined unary predicate St($x$), for the standard sets, and axioms relating the standard and nonstandard sets.
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| Tipo |
pdf
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| Palabras clave |
Filosofie Licentiatavhandling,Jonas Eliasson Ultrapowers as Sheaves on a Category of Ultrafilters
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| Tipo de Interactividad |
Expositivo
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| Nivel de Interactividad |
muy bajo
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| Audiencia |
Estudiante
Profesor
Autor
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| Estructura |
Atomic |
| Coste |
no
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| Copyright |
sí
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unrestricted |
| Formatos |
pdf
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| Requerimientos técnicos |
Browser: Any
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| Relación |
[IsBasedOn] http://www.math.uu.se/research/pub/FEliasson1.pdf
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| Fecha de contribución |
31-mar-2009 |
| Contacto |
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