The way that naïve set theory is usually presented in undergraduate education is via very concrete examples of sets, often involving non-mathematical elements. When power sets are treated, having a set as an element of some other set is just another new and exciting [YMMV] thing to consider. I saw a recent example that discussed how $\{\emptyset\}$ is both an element and a subset of $\{\emptyset,\{\emptyset\}\}$. To a traditionally trained set theorist this is perfectly natural. However, there are other axiomatisations of sets that do not allow sets to be elements of sets. In this setup the power set of $X$, which surely must have subsets of $X$ as elements, seems to be a little tricky to think of. But not really: one just has to realise that elements of the power set code, or are labels for, or name subsets of $X$.

I was wondering whether there has been any treatment at the level of introductory (and/or naïve) set theory that takes this tack, or if people have had any experience using such a trick in introducing the power set.

  • $\begingroup$ To me it doesn't seem plausible to say that there are axiomatizations of sets where you can't have sets be elements of other sets. For example, if you define a circle to be a set of points, then you are not allowed to have sets of circles? The link that you gave seems to suggest that subsets are viewed as morphisms which are (or correspond to) characteristic functions. Surely you can have sets of characteristic functions. $\endgroup$ Sep 12 '17 at 10:39
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    $\begingroup$ @John trust me, there is more than one first-order axiomatisation of sets (that is, using no prior concept of "set") in which no set is an element of another set. The one discussed at the blog post (which dates back to the 1960s) is the best studied perhaps. As to the problem you mention, say you have a "collection" (whatever that means) of functions $f_i: X\to Y$ indexed by a set I. Then you can write this as a single function $f: X\times I \to Y\times I$ with certain properties. At no point do I have a set whose elements are functions. $\endgroup$ Sep 12 '17 at 11:58
  • $\begingroup$ Maybe I misunderstood the paper from that blog, but it identifies subsets of $X$ with functions from $X$ to $2$ and throws in an axiom (axiom 6: For all sets $X$ and $Y$ there exists a function set from $X$ to $Y$) for the existence of function sets. The paper discusses power sets and uses the notation $2^S$ for the power set, understood to be the function set whose existence is given by axiom 6. If this isn't a set whose elements are functions, what is it? Axiom 6 calls it a set. Anyway, the book "Conceptual Mathematics: A First Introduction to Categories" might be what you are looking for. $\endgroup$ Sep 12 '17 at 14:09
  • $\begingroup$ Thank you for an interesting link. Personally though while I have no trouble imagining that you can build a version of set theory where sets can't contain other sets, I wonder what the point of that is for naive set theory. It seems that when talking about naive set theory sets being members of other sets is extremely intuitive. I can see how sets being the only object all together might seem strange but then can't you just have what most students really think of when thinking naive set theory. I.e. set theory with lot's of atoms? $\endgroup$
    – DRF
    Sep 12 '17 at 20:13
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    $\begingroup$ That explains it in part, although I am still skeptical that it would be helpful to ban sets which have sets as members. In probability theory, an event is a (measurable) subset of a sample space. Why should it be problematic to talk about the set of all events which are independent of a given event? I see little motivation for adopting a foundational approach which doesn't allow me to e.g. talk about sets of events. Your use of the phrase "workaround" suggests that you agree that this is at the very least a problem with such foundations. Still, I'll try to keep an open mind. $\endgroup$ Sep 13 '17 at 14:33

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