MoL201529: Zwart, Maaike Annebeth (2015) Sheaf Models for Intuitionistic NonStandard Arithmetic. [Report]

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Abstract
The aim of this thesis is twofold. Firstly, to find and analyse models for nonstandard natural arithmetic in a category of sheaves on a site. Secondly, to give an introduction in this area of research. In the introduction we take the reader from the basics of category theory to sheaves and sheaf semantics. We purely focus on the category theory needed for sheaf models of nonstandard arithmetic. To keep the introduction as brief as possible while still serving its purpose, we give numerous examples but refer to the standard literature for proofs. In the remainder of the thesis, we present two sheaf models for intuitionistic nonstandard arithmetic. Our sheaf models are inspired by the model I. Moerdijk describes in A model for intuitionistic nonstandard arithmetic. The first model we construct is a sheaf in the category of sheaves over a very elementary site. The category of this site is a poset of the infinite subsets of the natural numbers. Apart from the Peano axioms, our sheaf models the nonstandard principles overspill, underspill, transfer, idealisation and realisation. Many of our results depend on a classical metatheory. Moerdijk’s proofs are fully constructive, which is why we improve our site for our second model. For the second model, we use a site with more structure. In the category of sheaves on this second site, we find a nonstandard model that much resembles our first model. We get the same results for this model and are able to prove some of the results that previously needed classical metatheory, constructively. However, there remain principles of which we can only show validity in our model using classical logic in the metatheory. Lastly, we try to construct a nonstandard model using a categorical version of the ultrafilter construction on the natural numbers object of the category of sheaves on our first site. This yields a sheaf which has both the natural numbers object and our first model as subsheaves.
Item Type:  Report 

Report Nr:  MoL201529 
Series Name:  Master of Logic Thesis (MoL) Series 
Year:  2015 
Uncontrolled Keywords:  logic, mathematics 
Subjects:  Logic 
Date Deposited:  12 Oct 2016 14:39 
Last Modified:  12 Oct 2016 14:39 
URI:  https://eprints.illc.uva.nl/id/eprint/970 
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