Two ways of defining rank of a set

Question

I am studying set theory, and I have some difficulties in understanding how people define the notion of rank there (I hope, specialists in logic will excuse me for this).

As far as I understand, there are two equivalent ways of defining rank of a set:

1. Krzysztof Ciesielski in his book Set Theory for the Working Mathematician defines rank by the formula $$ \text{rank}(X)=\min\{A\in\text{Ordinal numbers}: \ X\in V_{A+1}\}, $$ where $V_A$ is what is called cumulative hierarchy.

2. J.Donald Monk in his Introduction to Set Theory defines rank by the formula $$ \text{rank}(X)=\min\{A\in\text{Ordinal numbers}: \ \forall Y\in X\quad \text{rank}(Y)< A\}. $$

There is no problem for me with the first definition, but I don't understand the second one.

J.D.Monk writes that his definition is justified by the

General recursion principle: each function $F:V\to V$ (where $V$ is the class of all sets) defines a unique function $G:V\to V$ by the formula $$ G(X)=F(G\big|_X),\qquad X\in V $$ (here $G\big|_X$ is the restriction of $G$ on $X$; I simplify a bit Monk's Theorem 13.1).

The problem for me is that I don't understand, which function $F:V\to V$ in these terms defines rank. I would think that Monk has in mind the function $$ F(H)=\min\{A\in\text{Ordinal numbers}: \ \text{Range}(H)\subseteq A\}. $$ But this function is not defined for all $H\in V$, only for those $H$ which have range in the class of all ordinals (I wrote this in one of my previous questions, here).

I suppose, there must be a standard trick, that people use here, but I don't know it. Can anybody clarify me this?

Answer 1

Just define $$F(H)=\min\{A\in\text{Ordinal numbers}: \ \text{Range}(H)\subseteq A\}$$ if $H$ is a function and every value of $H$ is an ordinal number, and $F(H)=\emptyset$ otherwise. By the general recursion principle, you then get a function $G$, and you can prove by $\in$-induction that $G(X)$ is an ordinal for all $X$ and so $G(X)$ is actually always given by the first case in the definition of $F$.

Answer 2

Long comment

We have to verify the conditions of Th.13.1.

1) The relation $\in$ is well–founded. and

2) The field of $\in$ is $V$ and, for all $x \in V$ (i.e.$x \in Fld (\in)$): $\{ y : y∈x \}$ is a set.

Now we have to define the function $F$ with domain $Fld (\in) \times V$, i.e. $V \times V$ such that:

$$F(x,u) = \min(\{ \alpha \in OR : u(y) < \alpha, \text { for each }y \in x \})$$

if $u$ is a function whose range is contained in $OR$ and $y = \emptyset$ otherwise.

Then, by recursion, there is a unique function $G$ such that $Dom(G) = Fld(\in)=V$ and for all $x \in V$,

$$Gx = F(x,G|_{\{ y : y \in x \}})$$

$$ρ(x) = \min(\{ \alpha : \rho(y) < \alpha, \text { for each } y \in x \}).$$