Are statements allowed to assert something about their containing sets?

Question

Let $X$ be a set.

Let $x \in X$.

Let $x$ be a statement $P(X)$ that asserts something about $X$.

I thought I read somewhere this isn't allowed. However I can't find anything on Google. Maybe I'm searching for the wrong things.

Answer 1

I suspect you might be thinking of the axiom of foundation. This axiom rules out, among other things, sets containing themselves as elements or even as subsets of elements. It is an axiom of ZFC, but there are set theories which don't include it (or even include outright anti-foundation axioms).

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On the other hand, if you really are asking about sentences:

I think this will become less mysterious if we focus instead on cooking up a sentence which refers to a set which happens to contain it; that is, build $x$ first and then find an appropriate $X$. For example:

$$\mbox{"The set of English sentences containing ten words is finite."}$$

Take "$x$" to be the sentence above, and "$X$" to be the set of English sentences containing ten words. While self-reference is always worrying, it's hard to find an objection to something this straightforward. Note in particular that there isn't really any (direct) self-reference going on here.

Now of course, ultimately what sets are "allowed" depends on exactly what set theory you're using. E.g. ZFC proves that there is no universal set, but NF proves that there is a universal set. Meanwhile, in both ZFC and NF every object is a set; sentences, as such, don't exist and are rather part of the metatheory (although they can be represented by sets in various ways). So if you're not satisfied with the above vague response, you have a lot of work to do to make your question precise.