Category Archives: Notes

Zornian Functional Analysis or: How I Learned to Stop Worrying and Love the Axiom of Choice

Back in the fall semester of 2015-2016 I had taken a course in functional analysis. One of the reasons I wanted to take that course (other than needing the credits to finish my Ph.D.) is that I was always curious about the functional analytic results related to the axiom of choice, and my functional analysis wasn’t strong enough to sift through these papers.

I was very happy when the professor, Matania Ben-Artzi, allowed me to write a final paper about the usage of the axiom of choice in the course, instead of taking an exam.

I have decided to finally post this paper online. It covers some possible disasters in functional analysis without the axiom of choice, or with “seemingly nice” assumptions (such as automatic continuity). You can find it in the Papers section.

My goal was to make something readable for analysts, rather than to provide a retread of older set theoretic proofs (and some model theoretic proofs). So some things are left with only a reference, and other set theoretic statements are formulated in a rather unusual way. If you are interested, I’d be happy to hear any remarks on this paper, or suggestions for improvements in the comments below or over email. If you know analysts that might be interested to read this, please let them know of the existence of the paper.

For the set theorists the paper can be seen as a nice historical overview of these results, and perhaps it can be of use in other ways.

The Five WH’s of Set Theory

I was asked to write a short introduction to set theory for the European Set Theory Society website. I attempted to give a short answer to what is set theory, why study it, when and how to study it and where to find resources.

You can find the article on the ESTS’ website “Resources” page, or in the Papers section of my website.

My goal was to present set theory in general, and $\ZF$-centric in particular. But I also included class set theories, atoms and non-well founded theories, as well as New Foundations, and an ending paragraph that points into additional directions (like categories or type theory) and philosophical questions (which should promptly be discussed over beer).

I initially tried to include references for $\ETCS$ or constructive set theories like $\mathsf{IZF}$ or $\mathsf{CZF}$. But I’m not familiar with references to these subjects, and the text began to grow longer than I had hoped. So those were omitted at the end.

In any case, feel free to leave me a comment, or an email, or a note attached to a beer, with your critique or opinion on this article.

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.


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

I assume that the reader is familiar with the basics of forcing, but the third chapter can be read even by those unfamiliar with forcing.

The Axiom of Choice and Self-Dual Vector Spaces

I have uploaded a note titled The Axiom of Choice and Self-Duality of Vector Spaces. Here is a short summary and background.

It is a well known fact (in $\ZFC$ at least) that if $V$ is a vector space, and $V^\ast$ is the algebraic dual of $V$ then $V\cong V^{\ast\ast}$ if and only if $\dim V<\infty$. Some long time ago, after a discussion with Pete L. Clark on math.SE, I set about finding a counterexample. After giving up at first, I found about automatic continuity. It turns out that in models like Solovay’s model every linear functional from a Banach space to the field (real numbers or complex numbers) is automatically continuous. In such model, if so, any reflexive Banach space is also isomorphic to its double dual.

I began writing a short note with all the relevant theorems and information, hoping to find my first publication there, but alas as I was finishing I saw that this result (and more) was already covered in Schechter’s immense book Handbook of analysis and its foundations. Now that I have a website, it seems like a good reason to make final adjustments and upload the note.

This is my first note, and any comment or suggestion will be most helpful for future notes (which are coming, I can assure you).


In models where every set of real numbers has the Baire property it turns out that every linear operator from a Banach space to a separable normed space is automatically continuous. In particular every linear functional is automatically continuous, and therefore the algebraic dual and the topological dual are the same.

Let $V$ be a vector space over $\mathbb R$. Denote by $V^\ast$ the algebraic dual of $V$, and for a topological vector space denote by $V^\prime$ the continuous dual. We will show that the following implications are not provable without the axiom of choice:

  1. $V\cong V^\ast$ implies that $\dim V<\infty$;
  2. $V\cong V^{\ast\ast}$ by a natural isomorphism if and only if $\dim V<\infty$;
  3. If $V$ is a Banach space, $V^\prime$ is reflexive if and only if $V$ is reflexive;
  4. If $V$ is a reflexive Banach space, $W\subseteq V$ is a closed subspace, then $W$ is also reflexive;
  5. If $V^\prime$ is separable then $V$ is separable.

    The Axiom of Choice and Self-Duality of Vector Spaces