MoL-2013-07: On The Modal Logics of Some Set-Theoretic Constructions

MoL-2013-07: Inamdar, Tanmay C. (2013) On The Modal Logics of Some Set-Theoretic Constructions. [Report]

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In set theory, there are various transformations between models. In particular, forcing, inner models, and ultrapowers occupy a fundamental place in modern set theory. Each of these play a different role. For example, forcing and inner models are typically used to establish the consistency of statements and the consistency strength of statements, and ultrapowers are typically used to define various large cardinal notions, which play the role of a barometer for consistency strength of statements. Each of these techniques however, can be seen as a process for starting with one model of set theory, and obtaining another. Indeed, it is this aspect of these techniques that we are interested in in this thesis. Each such method of transforming models of set theory lends itself to analysis by the techniques of modal logic [Ham03, HL08], which is the general study of the logic of processes. It is a recent trend in set theory that research has focussed on these modal aspects of models of set theory. This is partly due to philosophical concerns, such as Hamkins’s multiverse view [Ham09, Ham11], Woodin’s conditional platonism [Woo04], Friedman’s inner model hypothesis [Fri06], but also due to mathematical concerns, such as to account for the curious fact that, in some sense, these techniques that we have mentioned are essentially the only known techniques that set theorists have to prove independence results. Concretely, if we fix a particular technique of model-transformation, we may reasonably ask of a given model of set theory questions of the following nature: “which statements are always true in all models that we shall construct by using this technique?”; “which statements can we always change the truth value of in any model that we shall construct by using this technique?” etc. Questions of the first sort are the topic of study of the area of set theory which is known as absoluteness, whereas questions of the second sort are the topic of study of the area of set theory known as resurrection. However, in both these cases, the questions we are asking talk about specific sentences in the language of set theory. That is, while the answers to these questions change depending on the type of model-transformation technique that we are considering, they are not purely questions about these techniques. In this thesis, we are (for the most part) not interested in this interplay between a model-transformation technique and sentences in the language of set theory, but instead, in the purely modal side of these techniques. That is, we are interested in understanding the general principles that are true of these techniques when they are seen as processes. As an example of the kind of questions that we shall concern ourselves with, consider: “If φ is a statement that is true in some model that we construct by using this technique, and ψ is another statement that is true in some model that we construct by using this technique, then is it the case that we can construct a model where both φ and ψ are true by using this technique?”, or “If φ is true of all models that we shall construct by using this technique, is φ already true?”. Note that the answers to these questions do not depend on what φ and ψ are, but only on the nature of these model-transformation techniques. These questions were first considered by Hamkins in [Ham03]. In particular, Hamkins showed that by interpreting the modal operator by “in all forcing extensions” and the ♦ operator by “in some forcing extension” one could interpret modal logic in set theory in a very natural way, and using this interpretation, study the technique of forcing through the modal lens. Hamkins used this interpretation to express certain forcing axioms known as maximality principles. These axioms were meant to capture the essence of models where a lot of forcing had already occurred, or to quote Hamkins, “anything forceable and not subsequently unforceable is true”, and relativisations of ‘forceable’ to specific types of forcing notions. It is easily seen that modal logic provides an elegant way of expressing these statements using the scheme ♦ φ φ. Hamkins also gave a lower bound of S4.2 for the modal logic that arises from forcing, the modal logic of forcing, in this paper. Hamkins’s work on maximality principles has had many follow ups, the earliest ones being [Lei04] and [HW05]. The first paper devoted entirely to the modal logic of forcing was [HL08]. In particular, they were able to show that the modal logic of forcing is S4.2. They also studied various generalisations of the modal logic of forcing, such as the modal logic of forcing with parameters, and developed some techniques which modularise the process of calculating the modal logics of set-theoretic constructions. In addition to this, in [HL08], various relativisations of modal logic of forcing were also considered. For example, if we fix a definable class of partial orders P, and a definition for it, we may interpret the operator as “in all forcing extensions obtained by forcing with a partial order in P” and the ♦ operator as “in some forcing extensions obtained by forcing with a partial order in P” and ask what the modal logic so obtained, denoted by MLP , is. This line of investigation is the main topic of study of [HLL], where for many natural classes P, upper and lower bounds are given for their modal logic. We continue this line of enquiry in this thesis. In particular, we take P to be the class of ccc-partial orders, and we study their corresponding modal logic, MLccc. We are able to improve the upper bound for MLccc which was obtained in [HL08]. In order to do this, we generalise the method found there from the case of a single ω1-tree to the case of an arbitrary finite number of ω1-trees. Along the way, we obtain a characterisation of Aronszajn trees to which a branch can be added by ccc forcing which is interesting in its own right, and which also raises some questions of independent interest. Another different direction that we pursue is that of looking at a different technique for relating models, namely that of taking definable-with-parameters inner models. The germs of this endeavour can be found in [HL13], where the modal logic of the relation of being a forcing ground 1 is studied. We are able to compute the exact modal logic of this relation, though this modal theory was not one which had been considered in this area before. We obtain this theory by adding an extra axiom to the well-studied modal theory S4.2 which captures the property of L, Godel’s constructible universe, being in a sense the minimal model of ZFC. Our proofs strongly rely on the results from [HL13].

Item Type: Report
Report Nr: MoL-2013-07
Series Name: Master of Logic Thesis (MoL) Series
Year: 2013
Date Deposited: 12 Oct 2016 14:38
Last Modified: 12 Oct 2016 14:38

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