Von Neumann Universe, difference between Ordinal and Successor Ordinal 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!
 A: 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 $\alpha$.
A: 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.
A: 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
