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Realizability Interpretations for Intuitionistic Set Theories

Dihoum, Eman Emhemed (2016) Realizability Interpretations for Intuitionistic Set Theories. PhD thesis, University of Leeds.

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Abstract

The present thesis investigates the validity of some interesting principles such as the Axiom of Choice, AC, in the general extensional realizability structure V(A) for an arbitrary applicative structure, A, generalising the result by Rathjen established for the specific realizability model V(K_1), the Fan Theorem, FT, and the principle of Bar Induction, BI, in the particular realizability structures over the Graph Model, V(P(omega)), and over the Scott D1 Model, V(D_infty), since, in the literature, little is known about these realizability models and most investigations are carried out in the realizability models built over Kleene's first and second models. After an introduction and some background material, given in the first two chapters, I introduce the notion of extensional realizability over an arbitrary applicative structure, A, and I show that variants of the axiom of choice hold in V(A). Next, the focus switches from considering the general realizability structure V(A) generated on an arbitrary applicative structure, A, to the specific realizability universes, V(D_infty) and V(P(omega)) to investigate some interesting properties including the validity of FT and BI in these universes. For the remainder of the thesis, a proof of the soundness of realizability with truth, as it leads to different applications than that without truth, for the theories CZF and CZF + REA, is given and an investigation of many choice principles is carried out in the truth realizability universe V*(A) for an arbitrary applicative structure, A.

Item Type: Thesis (PhD)
Keywords: Realizability, Intuitionistic theory, Choice principles, Brouwerian principles.
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds)
The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds)
The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds)
Identification Number/EthosID: uk.bl.ethos.713230
Depositing User: Mrs E. E. Dihoum
Date Deposited: 16 May 2017 12:40
Last Modified: 25 Jul 2018 09:54
URI: http://etheses.whiterose.ac.uk/id/eprint/17237

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