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