Notation for elements nested within elements of a set?

For the purposes of something I'm working on, I need to define and use notation referring to the elements of sets which are themselves elements of another set, to an arbitrary depth, and that only appear once.

I've seen the notation $$x \in_n X$$ used with multisets, to denote that the element $$x$$ has a multiplicity of $$n$$ in the multiset $$X$$, so I figure I could use $$\in_1^k$$ and then define it alongside the formula. It's just cumbersome to use language like "an element of an element of an element of..." to a depth of $$k$$. Even "$$x$$ appears exactly once in the nested elements of $$X$$ at a depth of $$k$$" may be hard to understand. Maybe, "$$x$$ is a $$k$$th-order sub-element of $$X$$ with a multiplicity of $$1$$"?

Is there a more concise way I can describe this? Or, even better, are there already notations or terms that would fit this purpose? Although I kind of like my last idea. Is that an intuitive phrasing?

• I have seen $x \in^2 y$ for $x \in z \in y$ (for some $z$) in the literature, alongside its analogues $\forall x \in^2 y$ (which is the same as $\forall z \in y \forall x \in z$) and $\exists x \in^2 y$ (i.e. $\exists z \in y \exists x \in z$). Recursion gives you a nice definition that should be pretty natural. Jul 10, 2019 at 10:50

Let $$\cup^0 X=X$$ and $$\cup^{n+1}X=\cup (\cup^nX).$$ The transitive closure of X is tr cl $$X=\cup_{0\le n\in \Bbb Z}(\cup^n X)....$$ (A set $$Y$$ is transitive iff $$\forall y\in Y \,(y\subset Y).$$ And tr cl $$X$$ is the $$\subset$$-smallest transitive $$Y$$ such that $$X\subset Y.)$$

For $$x\in$$ tr cl $$X$$ let $$d_X(x)$$ be the least $$n$$ such that $$x\in \cup^n X.$$ ($$d$$ for depth.) And let $$\text { Lev}_X(n)=\{x: d_X(x)=n\}.$$ (Lev for level.)

Note that if $$d_X(x)=n>0$$ then $$x$$ may belong to two or more members of Lev$$_X(n-1).$$

I have seen the notation $$\cup^n$$ in a textbook $$[1]$$, and tr cl $$X$$ is standard, although with stylistic variety, e.g. TrCl($$X$$), but to be safe it is probably best to give their definitions when you are writing.

$$[1].$$ Kunen, Kenneth. Set Theory : An Introduction To Independence Proofs.

If K is a collection of sets, then x in $$\cup$$K states that x is an element of one of the sets in K.

x in A in K states that x is an element of the set A of the collection K.

• This is not really helpful when I need to refer to elements an arbitrary number of layers deep, because it means I'd have to write (or imply) a bunch of labels like $A_1, A_2,\cdots,A_n$ that I'm never going to refer to again in the expression. And linguistically, it'd be even more unwieldy. Jul 10, 2019 at 10:26