Asaf Karagila
I don't have much choice...

Blog posts from 2013

Equivalent to the axiom of choice that I didn't know about

First I must apologize. I wanted to write a second post about forcing and preserving choice principles (I gave a nice talk in the student seminar about a week after the previous post), and I had a lot of things to say. I just ended up not writing it, and for absolutely no good reason. And somehow things continued that way and I felt more and more awkward to post anything because of that, but the vicious cycle must break somewhere.

I recently tried to figure out the consequence of some forcing in \(\ZF\). This has led me to the following statement:

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The cardinal trichotomy: finite, countable, and uncoutnable.

There is a special trichotomy for cardinality of sets. Sets are either finite, or countably infinite, or uncountable. It's an interesting distinction, and it has a very deep root -- at least in my perspective -- in the role of first-order logic.

Finite objects can be characterized in full using first-order logic. The fact that you can write down how many elements a set have, is a huge thing. For example, every finite structure of a first-order logic language has a categorical axiomatization. If the language is finite, then the axiomatization is finite as well.

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Strong chain conditions and preservation of choice principles

I recently returned from a wonderful week in Italy, where I attended the Young Set Theorist 2013 conference. I met a lot of new people, some old acquaintances, baffled people with oversized pickles, and most importantly shared and learned some great ideas.

One of the nicer things I'd done was to work with Thomas Johnstone on some preservation theorem related to forcing and choice principles (see also this announcement by Victoria Gitman). In order to clean up a bit the proof, I'll introduce a new definition which is going to slightly extend the ideas originally discussed in Italy. So without further jibber jabber, let's talk mathematics.

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Provable Equality Of Exponentiation

It's an almost trivial theorem of cardinal arithmetics in \(\ZF\) that given four cardinals, \(\frak p,q,r,s\) such that \(\frak p<q,\ r<s\) we have \(\frak p^r\leq q^s\).

In a recent question on math.SE some user has asked whether or not we always have a strict inequality. Everyone sufficiently familiar with the basics of independence results would know that it is consistent to have \(2^{\aleph_0}=2^{\aleph_1}=\aleph_2\), in which case taking \(\mathfrak{p=r}=\aleph_0,\ \mathfrak{q=s}=\aleph_1\) gives us equality. But it's also trivial to see that we can always pick cardinals whose difference is large enough to keep the inequality true.

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Choice Principles: What are they?

What does the phrase "\(\varphi\) is a choice principle" mean? This is something that I have spent quite a lot of my time thinking about. Directly and indirectly. What are choice principles as we know them? And who gets to decide?

For a set theorist, at least a "classical" set theorist (working within the confines of \(\ZF\) and its extensions to \(\ZFC\) and so on), a choice principle can aptly be defined as "Sentence \(\varphi\) in the language of set theory which is provable from \(\ZFC\) but independent from \(\ZF\)". Indeed that is how I think of choice principles, and how I referred to them in my masters thesis (albeit I prefaced that definition by pointing out its naivety).

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Infinite dimensions and the axiom of choice

In a recent math.SE question, Thomas Andrews asked whether or not the existence of an infinite linearly independent set in a vector space which is not finitely generated requires the axiom of choice.

The answer is positive. It does require the axiom of choice. The counterexample is due to Läuchli who constructed a model in which there was a vector space which was not finitely generated, but every proper subspace is finitely generated. Given such vector space it is obvious that no infinite set can be linearly independent.

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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.

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The Philosophy of Cardinality: Pathologies or not?

What are numbers? For the layman numbers are those things we use for counting and measuring. The complex numbers are on the edge of being numbers, but that's only because they are taught in high-schools and many people still consider them imaginary (despite them having some reasonably applicative uses).

But a mathematician knows that a number is basically a notion which represents a quantity. We have so many numbers that I don't even know where to begin if I wanted to list them. Luckily most of the readers (I suppose) are mathematicians and so I don't have to.

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On Leinster's "Rethinking set theory"

There has been a lot of recent discussions regarding Tom Leinster's paper "Rethinking set theory" (arXiv). Being an opinionated person, I only found it natural that I had an opinion on the paper. Now that I have a blog, I have a place to write this opinion as well.

The paper challenges the hegemony of \(\ZFC\) as the choice set theory. It offers an alternative in the form of \(\newcommand{\ETCS}{\axiom{ETCS}}\newcommand{\ETCSR}{\axiom{ETCS+R}}\ETCS\), a categories based set theory. The problem with \(\ETCS\) is that it is slightly weaker than \(\ZFC\). But we also know how much weaker: it lacks the expressibility of the full replacement schema. In this case we can just add a replacement schema-like list of axioms to have \(\ETCSR\).

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