When might identifying singleton subsets with their (single) elements "cause confusion"? I'm reading Givant's _Duality Theory for Boolean Algebras with Operators_ and on p. 14 he makes an off-hand remark when discussing how to correlate an algebra with a relational structure: > If we identify each singleton subset $\\{r\\}$ of $U$ with the element $r$ itself -- as is often done when no confusion can arise.... (p. 14) He then goes on to define the algebra's operations in terms of relations holding of sequences these elements. When might "confusion arise" in identifying elements with their singletons? Is he just alluding to this identification _not_ consisting in the adoption of a non-well-founded set theory like Quine's _New Foundations_ , where _urelements_ are identified with their singletons? Or is there a less exotic case that might arise that he could be worried about?

Well, identifying $r$ and $\\{r\\}$ involves pretending that two things which aren't equal (namely, $r$ and $\\{r\\}$) actually are equal, so you can use the same name for both of them. This obvious could cause confusion in any situation where it is important to distinguish them. For instance, imagine if you wanted to talk about the set $\\{r,\\{r\\}\\}$: try doing that while using the same symbol for $r$ and $\\{r\\}$! Or, imagine that there is an element $r\in U$ such that $\\{r\\}$ is also an element of $U$. Then identifying $r$ and $\\{r\\}$ could be quite confusing: when you refer to $r$, do you mean the element $r$ of $U$ or the element $\\{r\\}$?

Now these examples are somewhat contrived, and in most cases you care about there is no real danger of such confusion. But whenever you start saying things that are not literally true (like pretending $r$ and $\\{r\\}$ are the same), you should be alert to the possibility of such confusions.