Don't worry about itThere are 5 comments on this post.
Nov 09 2015, 16:37
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.
And it seems to me that a lot of the flak set theory gets comes from the inability, or rather the unwillingness of people outside of set theory to study just the little bit more than they absolutely have to. Perhaps set theory wasn't properly introduced to them in their undergrad, maybe too early and maybe too late (after they had heard bad things about it from their algebra teachers complaining about proper classes and the axiom of choice).
I don't know why people don't do that, and I am very happy to have had the opportunity to work with Azriel Levy in the past three years and see how so many of our students love the topic. And this is not because Israel is a friendly place towards set theory, this becomes increasingly less true. It's because they have a teacher and a teaching assistant (who is teaching another complementary half-course in the exercises) which come from set theory and do not shy away from questions like "Can we define ordered pairs in a different way? What happens then?" and it's because they have teachers which are truly enthusiastic about set theory.
Students are impressionable. If you have a good teacher and they tell you something, this will stick with you. And if that good teacher tells you that set theory is the root of all evil, then you will continue to think so until, if you're lucky, you'll learn otherwise. Or, if you have a good teacher that tells you in your sophomore year that the axiom of choice is a dormant research topic, but it has a lot of beautiful mathematics and a lot of interesting open questions left... well, you end up like me.
So what's the whole issue here? Is \(\pi\subseteq\RR\)? That depends, you haven't given me a definition for the sets \(\pi\) and \(\RR\). But once you will give me a definition, my answer is that it doesn't matter, because we are interested in structures with a certain property, not with specific sets. And that given any other way to interpret the real numbers as sets can yield different answers to the question.
And I truly don't understand why this bothers people. There are three paths from here to the nearest groceries store, and all take about the same time and effort to cross. It means that if you ask me to go to the groceries store you can ask me if I passed this or that place, and the answer will depend on which path I took. And thank goodness I am not afraid of questions like "is \(\pi\subseteq\RR\)?", because then the choice of path to the store would have terrified me, and I would be forced to move to the building right next to it, which I am told is really shitty.
In short, if there's one thing to take from this post is that the people who are terrified of the ability to make sense of the questions like "is \(\pi\subseteq\RR\)?" are the people who are too terrified to understand that even set theorists don't care about these questions. And if you worry about them, and find that to be a good argument against \(\ZF\), then don't. There are reasonable arguments against \(\ZF\), but this is not one of them.