Von Neumann Universe, difference between Ordinal and Successor Ordinal

Question

I'm having some difficulty understanding the difference between two ways that ordinals appear to be defined. One way is an ordinal $\beta$ is defined by:

$V_\beta := \mathcal{P}(V_\beta)$

And results in the following ranked ordinals:

$V_0 := \emptyset = \emptyset$
$V_1 := \{0\} = \{\emptyset\} $
$V_2 := \{0, 1\} = \{\emptyset, \{\emptyset\}\} $

Then the successor ordinal $\alpha$ is defined slightly different (no power set) by:

$S(\alpha) = \alpha \space \cup \{\alpha\}$

Can anyone explain the difference between $V_\beta$ and $S(\alpha)$? I would expect the successor to be the next ranked ordinal $V_\beta$, however after expanding $V_3$ these two definitions are not equal:

$V_{3a} = S(V_2) = \{0, 1, 2\} = \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}$

while:

$V_{3b} = \mathcal{P}(V_2) = \{0, 1, 2, ?\} = \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\}$

Note: $V_{3b} \backslash V_{3a} = \{\{\emptyset\}\} $

I'm hoping to get this down as it seems like it forms a basis for future learning. Any insight appreciated. Thanks!

Answer 1

The $V_\alpha$ are not ordinals. They are the ranks of the Von Neumann hierarchy, which are indexed by the ordinals.

However, they do have a similarity. The ordinal $\alpha$ is the set of all ordinals less than $\alpha$. $V_\alpha$ is the set of all sets with rank less than $\alpha$.

Answer 2

The sets $V_\beta$ are not ordinals. They are a family of sets indexed by ordinals: we have one set $V_\beta$ for each ordinal $\beta$. As such, they are not an "alternate definition of ordinals" but are instead just certain sets that are related to but quite distinct from ordinals.

Answer 3

Your first equation should be $V_{\beta+1} := \mathcal{P}(V_\beta)$, but that is not for defining ordinals. It is for defining levels of the cumulative hierarchy, which are indexed by ordinals, the subscipts. $V_\beta$ is not the same thing as $\beta$. Your definition of a successor ordinal is the usual one. Your $V_{3a}$ is the usual set denoted by $3$, but is not a stage of the cumulative hierarchy. Both of your definitions only work for successor ordinals. You need a definition for limit ordinals. The usual definitions are $V_\beta = \bigcup_{\alpha \lt \beta} V_\alpha$ for $\beta$ a limit and $\beta = \bigcup_{\alpha \lt \beta} \alpha$ for $\beta$ a limit