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

Posts tagged structural set theory

In praise of Replacement

I have often seen people complain about Replacement axioms. For example, this MathOverflow question, or this one, or that one, and also this one. This technical-looking schema of axioms state that if \(\varphi\) defines a function on a set \(x\), then the image of \(x\) under that function is a set. And this axiom schema is a powerhouse! It is one of the three component that give \(\ZF\) its power (the others being power set and infinity, of course).

You'd think that people in category theory would like it, from a foundational point of view, it literally tells you that functions exists if you can define them! And category theory is all about the functions (yes, I know it's not, but I'm trying to make a point).

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Don't worry about it

In a recent Math.SE question about the foundations of category theory without set theory, someone made a claim that \(\ZF\) makes it hard to learn mathematics, because in \(\ZF\) the questions "is \(\RR\subseteq\pi\)?" and "is \(\RR\in\pi\)?" can be phrased. They continued to argue that there are questions like whether or not hom-sets are disjoint or not, which are hard to explain to people who are "drunk on ZF's kool-aid".

So I raised a question in the comment, and got replies from two other people who kept repeating the age old silly arguments of what are the elements of \(\RR\times\RR\) or what are these or that elements. And supposedly the correct pedagogical answer is "It does not matter what are the elements of \(\RR\times\RR\)." With that I strongly agree, and when I taught my students about ordered pairs on the very first class of the semester, I made it very clear that there are other ways to define ordered pairs and that we only do that because we want to show that there is at least one way in which ordered pairs can be realized as sets; but ultimately we couldn't care less about what way they encode ordered pairs into sets, as long it is a "legal" way.

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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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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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