# Definition talk:Von Neumann Hierarchy

I have changed the notation from $V_x$ to $V \left({x}\right)$ as the former is nonstandard and inferior. Also note it remains undocumented on ProofWiki. --prime mover (talk) 06:48, 23 August 2012 (UTC)

OK, but note that, once cardinality is developed a little more, the aleph function is usually denoted $\aleph_\alpha$ not $\aleph ( \alpha )$. Also, the fact that its domain is the ordinals is justified directly from Transfinite Recursion/Theorem 2. I'm not sure how to state this. --Andrew Salmon (talk) 06:51, 23 August 2012 (UTC)
$\aleph_\alpha$ as may be, but this is a special non-standard case (a bit like $n^+$ for the successor mapping) and, as I pointed out, not been defined on ProofWiki.
And I'm not sure that "its domain is the ordinals is justified directly from Transfinite Recursion/Theorem 2" does completely: the assumption that $V \left({\mathcal P \left({x}\right)}\right)$ is an ordinal is not obvious. How to state it? Write a page proving it. --prime mover (talk) 07:03, 23 August 2012 (UTC)
$\mathcal P ( V ( x ) )$ need not be an ordinal--that's part of its range, not its domain. Its definition is pretty straightfoward from Transfinite Recursion/Theorem 2 (just set $H$ to the powerset function and $G$ to that complicated class to remove those hypotheses) and the fact that its domain is $\operatorname{On}$ is a specific consequence of that definition. --Andrew Salmon (talk) 07:08, 23 August 2012 (UTC)

Agreed with Asalmon. There is nothing wrong with this definition. --Lord_Farin (talk) 09:27, 23 August 2012 (UTC)

Yes of course. As it originally stood, the codomain of $V$ was not made clear. It's better now. --prime mover (talk) 20:45, 23 August 2012 (UTC)