Vector Spaces and Antichains of Cardinals in Models of Set Theory

I finally uploaded my M.Sc. thesis titled “Vector Spaces and Antichains of Cardinals in Models of Set Theory”.

There are several changed from the printed and submitted version, but those are minor. The Papers page lists them.

Abstract:

Läuchli constructed a model of \(\ZF\) in which there is a vector space which is not of finite dimension, but every proper subspace is of a finite dimension. In Läuchli's model the axiom of choice fails completely, there is a countable family from which we cannot choose representatives.

In this work we generalize Läuchli's original proof. In the proof presented here we show that we may choose any cardinal \(\mu\) and construct a model of \(\ZF\) in which there is a vector space such that every proper subspace has dimension less than \(\mu\), but the vector space itself is not spanned by any linearly independent subset. The construction uses a technique called symmetric extensions, which is used to create models in which the axiom of choice fails. In the first chapter we will review this technique, and weak versions of the axiom of choice. We show that in our construction we may preserve relatively large fragments of choice in the universe.

We also generalize a theorem by Monro which states that it is consistent without the axiom of choice that there are infinite sets which have no countably infinite subset, but can be mapped onto very large ordinals. Our proof uses the method of symmetric extensions, in contrast to Monro which took a different approach, and we show that for any two regular cardinals \(\lambda\leq\kappa\) we may construct a model of \(\ZF\) in which there is a set that can be mapped onto \(\kappa\), and \(\lambda\) is the least ordinal which cannot be injected into this set.

In the third chapter we present a recent paper of Feldman, Orhon and Blass. In this paper the authors prove that if there is a finite bound on the size of antichains of cardinals then the axiom of choice holds. We review the original results and extend them to hold for a weaker notion of a quasi-ordering of the cardinals. We also answer one of the questions presented in the paper, and add questions of our own.