I don't understand the argument of how the union of the set of all singletons results in a set of all sets. Doesn't the union of a set of all singletons just contain one set - a set of all possible elements? And not multiple sets? This is how I imagine it: $$ \bigcup \{\{a\}, \{b\}, \{c\}, \dots\} = \{\{a, b, c, \dots\}\}$$
What am I missing?
In three words: sets ARE elements. So the two notions are equivalent. Natural numbers, rationals, and reals are just special sets. See here for more info. (This answer hides a lot of the dirty details - you can have a set theory with urelements as they're called, but within ZFC there are no such objects.)
For example, $\{\{\}\}$ is a singleton - it contains exactly one element, which is the empty set, and so the empty set $\{\}$ is contained in your union. The same is true for any other set $A$.
