MoL-2006-03: Strong limits and Inaccessibility with non-wellorderable powersets

MoL-2006-03: Dimitriou, Ioanna (2006) Strong limits and Inaccessibility with non-wellorderable powersets. [Report]

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Abstract

This thesis is about set theory without the axiom of choice. The theory of ordinals and their powersets without the axiom of choice is not a popular subject in set theoretic practise; in this thesis, we will shed a little light on some basic questions in this area. Our basic theory is ZF, unless otherwise stated. In this thesis we will be interested in the concept of a cardinal being a strong limit. This concept is one of the basic properties of the ordinary theory of cardinals and their powersets. It is well studied in the ZFC context and is typically defined as follows: \kappa is a strong limit IFF(def) \forall \lambda<\kappa (2^\lambda<\kappa) where we read "2^\lambda<\kappa" as \some (any) ordinal in bijection with the powerset of \lambda is smaller than \kappa". In the ZFC context, this ordinal always exists, if 2^\lambda is not wellorderable, it may not. As it turns out, this definition is equivalent in ZFC to four other definitions (where < is replaced by relations <_s, <_i, <_{s-} <_{i-}) that are more appropriate for an investigation without the axiom of choice. We look at this subject from two different points of view, thus this thesis includes two parts. The first part is looking at the problem from an axiomatic point of view, i.e., we see what different answers we can have when we assume different axioms. It starts with the axiom of choice, the axiom of determinacy, weaker forms of them which are involved in this study and some generalisations of statements incompatible with the more famous axioms above. We end this part with a discussion on several notions of being an inaccessible cardinal, i.e., a regular strong limit cardinal. These are defined using the alternative definitions of strong-limitedness we mentioned above. We also define the notion of being a \beta-inaccessible cardinal that uses the set of ultrafilters on a cardinal and it is connected with the axiom of determinacy. The notion of inaccessibility is connected with a metamathematical point of view in set theory. It is known that the existence of inaccessible cardinals is equivalent to ZFC having a set model. By Goedel's Incompleteness and Completeness Theorems this is actually a metamathematical proof that these cardinals' existence cannot be proven in ZFC. From this metamathematical point of view, theorems that talk about the consistency of a theory motivate us to define a consistency strength hierarchy between theories that contain ZF. This is because we conventionally accept ZF as consistent. We will not go into the details of this hierarchy but we will just state that a theory T has stronger consistency strength than a theory T' if T can prove the consistency of the same or more theories than T' can. Therefore the theory ZFC+"there is an inaccessible cardinal" is stronger than ZFC. All this creates the natural question of what happens in non-AC environments. This leads us at the second part of the thesis, where we look at the problem by constructing generic models by forcing. First we take a brief look at a model by Blass where all ultrafilters are principal and where the notion of being a \\beta-strong limit becomes trivial. Afterwards we will describe the method of taking symmetric submodels of generic extensions and then we will study in depth the Feferman-Levy model, a symmetric submodel. This model will answer most of our questions in this part and this will lead us to attempt a generalisation of it, hoping this will solve our last question. This attempt will fail but this failure will make the problem clearer and might help to lead us in a new way to approach this in the future.

Item Type: Report
Report Nr: MoL-2006-03
Series Name: Master of Logic Thesis (MoL) Series
Year: 2006
Uncontrolled Keywords: strong limits, inaccessible cardinals, consistency strength, set theory without the axiom of choice
Date Deposited: 12 Oct 2016 14:38
Last Modified: 12 Oct 2016 14:38
URI: https://eprints.illc.uva.nl/id/eprint/766

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