Three definitions of 'singleton set'? I discovered that there are three definitions of 'singleton set', and that these are at different levels of the set hierarchy.
A singleton set...

*

*(element level) ...has exactly one element;

*(set level) ...has exactly one strict subset (viz. the empty set);

*(family level) ...is an element of every family that covers it.

(Here "F covers A" means "F 's union equals A".  Perhaps this is not official terminology.)
My quite vague question: It seems there might be a bigger story behind these different ways of defining this same concept?
 A: Imagine you are a child or an AI robot with an incredible intelligence. You become fascinated and amused by informally thinking about (with no references) the finite symmetric groups $S_n$.
Eventually you want to formalize this 'slice of math', and attempt to layout a formal theory. You already understand how to construct the finite von Neumann ordinals,
0   = {}           = ∅
1   = {0}          = {∅}
2   = {0, 1}       = {∅, {∅}}
3   = {0, 1, 2}    = {∅, {∅}, {∅, {∅}}}
4   = {0, 1, 2, 3} = {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}
etc.

and regard these sets as canonical.
You decide that each of these collections of automorphisms must have an identity and begin by explicitly constructing $S_1$. Using recursion, you know that with $S_n$ defined you can construct $S_{\sigma(n)}$ where $\sigma(n)$ is the next ordinal.
So you've constructed a chain of proper natural inclusions,
$\quad S_1 \hookrightarrow S_2 \hookrightarrow S_3 \hookrightarrow \dots $
You develop your theory further and note that
$\;$ There is one and only one group structure on a singleton set.
$\;$ There is one and only one homomorphism of $S_1$ into $S_n$.
$\;$ There is one and only one homomorphism of $S_n$ into $S_1$.
Just for fun you decide to postulate the following as an axiom,
$\; \text{There exist a group } S_\omega \text{ such that for every } x \in S_\omega \text{ there exists an ordinal } n \text{ with } x \in S_n$
finding no contradictions and concluding that $S_\omega$ must be unique.
You also observe that there is one and only one way to re-frame a singleton set as a pointed set.
Having studied philosophy, you recall the quote

A journey of a thousand miles must begin with a single step.
Lao Tzu

A: Here is an interesting recast of the OP's family level definition.
Recall the definition of a partition refinement.
The following are true:
$\;$ The coarsest partition of a nonempty set is a singleton set.
$\;$ Every block in the finest partition of a set is a singleton set.
$\;$ A nonempty set is a singleton if and only if it has exactly one partition (finest = coarsest).
This is very elementary; it doesn't even require the formulation of an ordered pair.
In the next section we copy an extract from the Bulletin of Symbolic Logic.
Going back further before the advent of set theory, you'll find Gottfried Leibniz's Monadology philosophy. In today's mathematics if you have a singleton then it contains a single element that is also a set. By the above, that set can be partitioned into singletons. Is their a monad (or urelement) anywhere in our future?
In the last section we copy out an abstract from Springer Link.

The Empty Set, The Singleton, and the Ordered Pair
Akihiro Kanamori
Department of Mathematics, Boston University
For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice are broached in the formal elaboration of the ‘set of’f {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as amotif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the iterative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all.

Published: 18 June 2011
Monads and Mathematics: Gödel and Husserl
Richard Tieszen (1951-2017)
Department of Philosophy, San José State University
Abstract
In 1928 Edmund Husserl wrote that “The ideal of the future is essentially that of phenomenologically based (“philosophical”) sciences, in unitary relation to an absolute theory of monads” (“Phenomenology”, Encyclopedia Britannica draft) There are references to phenomenological monadology in various writings of Husserl. Kurt Gödel began to study Husserl’s work in 1959. On the basis of his later discussions with Gödel, Hao Wang tells us that “Gödel’s own main aim in philosophy was to develop metaphysics—specifically, something like the monadology of Leibniz transformed into exact theory—with the help of phenomenology.” (A Logical Journey: From Gödel to Philosophy, p. 166) In the Cartesian Meditations and other works Husserl identifies ‘monads’ (in his sense) with ‘transcendental egos in their full concreteness’. In this paper I explore some prospects for a Gödelian monadology that result from this identification, with reference to texts of Gödel and to aspects of Leibniz’s original monadology.
