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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!

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  • $\begingroup$ We prove there exists a $cardinal$ ordinal larger than the infinite cardinal ordinal $A$ as follows: Let $S$ be the set of members of $P(A\times A)$ that are the (graphs of ) well-orderings. For each $\sigma \in S$ there is a $unique$ ordinal $B(\sigma)$ isomorphic to $\sigma.$ By Replacement we have the set $C=\{B(\sigma): \sigma \in S\}.$ Then $\{c\in C: |c|\leq A\}$ is equal to $A^+,$ the least cardinal ordinal greater than $A.$ $\endgroup$ – DanielWainfleet Aug 25 '17 at 8:50
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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 $\alpha$.

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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.

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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

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  • $\begingroup$ I think the first equation is in Computerese. Programmers write stuff like $x:=x+1$. $\endgroup$ – DanielWainfleet Aug 25 '17 at 8:37

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