MoL-2012-10: Supremum in the Lattice of Interpretability

MoL-2012-10: Henk, Paula (2012) Supremum in the Lattice of Interpretability. [Report]

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Abstract

This thesis is located in the field of provability and interpretability logic, where modal logic is used in the study of formal systems of arithmetic. The central notion of this thesis is that of interpretability. The notion of interpretability can be seen as a tool for comparing axiomatic theories. Intuitively, if a theory T interprets a theory S , T is at least as strong as S. The modal logic ILM captures exactly what Peano Arithemtic (PA) can prove about interpretability between nite extensions of itself. As it turns out, nite extensions of PA form a lattice under the relation of interpretability, i.e. any two theories have an inmum and a supremum in the interpretability ordering. The supremum in this lattice is the main subject of study in this thesis. We will extend the logic ILM with a binary operator for the supremum, and explore the possibilities of having a modal semantics for the resulting system ILMS. For that purpose, the supremum will be studied both from the arithmetical as well as from the modal perspective. First, we will see that the exact content of the logic ILMS depends on the formula that is chosen as the arithmetical representative of the supremum. This is dierent from ILM, where the meaning of the modal symbols is xed from the outset. Proceeding to the modal side, we establish an important negative result: there can be no structural characterization of ILM{models that validate the dening axiom for the supremum. This precludes the possibility of having a relational semantics for the system ILMS | at least one that would extend the usual semantics for ILM. Finally, we examine an elegant but unfortunately failed attempt to nd a relational semantics for a particular representative of the supremum.

Item Type: Report
Report Nr: MoL-2012-10
Series Name: Master of Logic Thesis (MoL) Series
Year: 2012
Uncontrolled Keywords: Logic, Mathematics
Depositing User: Tanja Kassenaar
Date Deposited: 12 Oct 2016 14:38
Last Modified: 12 Oct 2016 14:38
URI: https://eprints.illc.uva.nl/id/eprint/876

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