MoL-2013-07:
Inamdar, Tanmay C.
(2013)
*On The Modal Logics of Some Set-Theoretic Constructions.*
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## Abstract

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 |

URI: | https://eprints.illc.uva.nl/id/eprint/896 |

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