I've just identified that the definition we used for the rank of a set in my set-theory class (1.) is different than the one I commonly find on the web (2.).
- $\text{rank}(A)=\min\{\alpha\mid A\in{V}_\alpha\}$
- $\text{rank}(A)=\min\{\alpha\mid A\subseteq{V}_\alpha\}$
There are a couple of key differences.
Empty set:
- $\emptyset\in\{\emptyset\}={V}_1\implies\text{rank}(\emptyset)=1$
- $\emptyset\subseteq\emptyset={V}_0\implies\text{rank}(\emptyset)=0$
Ordinals:
- $\alpha\in{V}_{\alpha+1}\implies\text{rank}(\alpha)=\alpha+1$
- $\alpha\subseteq{V}_\alpha\implies\text{rank}(\alpha)=\alpha$
Usefulness in proofs:
- $\text{rank}(A)$ is always a successor ordinal
- $\not\exists{A}\left(\text{rank}(A)=\omega\right)$
- $\not\exists{A}\left(\lim\left(\text{rank}(A)\right)\right)$
- $\text{rank}(A)$ is the smallest ordinal greater than the rank of every member of $A$
- $\forall\alpha\exists{A}\left(\text{rank}(A)=\alpha\right)$
So here's what I'm wondering:
- How common is the first definition (the less common one)?
- Are there any other advantages or subtleties in either that I'm missing?