Defining the smallest subset of $\Pi$ that's closed under $f$ and contains $a$ I need to model family relationships so I want to define the set of a person's direct descendants. I try to use recursion to construct such a set but I'm not sure if my definition is just circular.
My definition is as follows:

Let $\Pi$ be the set of all people. For $x \in \Pi$, their father is given by $F(x): \Pi \to \Pi$. Then, for any male, $\alpha \in \Pi$, the set of their direct descendants is given by
$$\mathcal F_{\alpha} = \{x \in \Pi: F(x) \in \mathcal F_{\alpha} \wedge \alpha \in \mathcal F_{\alpha} \}$$

I'm also considering

$$\mathcal F_{\alpha} = \{x \in \Pi: F(x) \in \mathcal F_{\alpha}\} \cup \{\alpha\}$$

Are these definitions valid? Are they equivalent? Do they accomplish my goal?
Let me know what you guys think. Thanks!
 A: You should generally be wary of using the object being defined in a definition of that object. Really, such an expression isn't really a definition at all. Rather, it should be thought of as a kind of "logical equation" which - hopefully!! - has a unique solution, and that more complicated expression ("the unique solution of ---") is a valid definition ... once a correct proof of unique-solution-ness has been exhibited.
Neither of your candidates works in this respect. For example, besides the intended interpretation, the choice $X=\Pi$ also satisfies the equation $$X=\{x\in \Pi: \mathcal{F}(x)\in X\wedge x\in X\}.$$
By contrast, the logical equation $$Z=\{z: z\in Z\rightarrow z\not\in Z\},$$ despite looking very weird at first glance, does have exactly one solution (namely $Z=\emptyset$); as yet another contrasting example, the equation $$A=\{x: x\not\in A\}$$ has no solutions at all.

OK, so what do you do?
Well, one option is to build on what you've already done. When you have an intuitively-compelling logical equation with too many solutions, one possible fix is to see if the intended solution is in some way "distinguished." In particular, you may look for the intersection of all solutions, e.g. $$Y=\bigcap_{X=\{x\in \Pi:\mathcal{F}(x)\in X\wedge x\in X\}} X.$$ Even if - as in your case - the equation you have in mind has too many solutions, this object is uniquely defined, and moreover is (as long as your equation has a certain form, which yours does) also a solution to your equation - the smallest solution, in a precise sense.
Another option is to work with sequences. Think about buildng the relation "$b$ is a descendant of $a$" in stages, with stage $n$ being the relation "$b$ is a descendant by $\le n$ generations of $a$." You get a sequence of relations $(R_i)_{i\in\mathbb{N}}$, and the relation you want is the union of this sequence of relations: $b$ is a descendant of $a$ iff $bR_ia$ for some $i\in\mathbb{N}$. This is a "from-below" construction, as opposed to the "from-above" construction in the previous paragraph; each approach has its place, and understanding why they give the same thing is a fundamental exercise!
