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04-25-2014 Enriched with more details

Let define the graphs $(V_\alpha,\in_\alpha)$ in $ZFC$

$V_\alpha$ can be finite too

$\in_\alpha \subseteq V_\alpha \times V_\alpha$ and $\in_\alpha:=\{(a,b):a \in b\}$

note 1: I'm very curious about the links betwen this collection of graphs that I'm describing and the axioms that they satisfies, if I'm not wrong they satisfie $ZFC$ only if $\alpha$ is inaccesible and for $\alpha=\omega$ we have a model of sets without axiomm of infinity. but what we have for smaller ordinals or finite?

$Q1.$ Do these structures have a special name as graphs? And what is the deep link with set theory?

I tried to draw the graphs of the first 4 cases $V_1$, $V_2$, $V_3$ and $V_4$ and I noticed that the nubers of edges of each graph are $$|\in_{\alpha+1}|=\sum_{a \in V_{\alpha+1} }|a| $$

note 2: For finite ordinals the thing is more interesting since it is always the sum of ${^\alpha}2-1$ non-empty sets (where $^{n}a$ is tetration) we "should" have

$$|V_{\alpha+1}\setminus \varnothing|={^\alpha}2-1 \leq|\in_\alpha|$$

even if I don't know if it holds always (I mean for infinite ordinals) but the best I could find was the following...but i'm not sure is correct

$$|\in_{\alpha+2}|=\sum_{i=1 }^{{^\alpha}2}i({^\alpha}2-i+1) $$

$Q2.$ How I can find a closed form for $|\in_{\alpha}| $ and $\alpha<\omega$? What is the pattern for infinite ordinals, what is the upper bound of $|\in_\alpha|$?

I noticed that in the graphs $(V_\alpha,\in_\alpha)$ we can define the cardinality and some other concepts of set theory but limited to $V_\alpha$ and an interesting concept of "inverse cardinality" too.

Elements of a node are the "predecessors" $$elem_{\in_\alpha}(a):=\{b: b\in_\alpha a\}$$

and the cardinality of a node $$|a|_{\in_\alpha}=|elem_{\in_\alpha}(a)|$$

The "owners" of a node are the "successors" $$own_{\in_\alpha}(a):=\{b: a\in_\alpha b\}$$

now my idea ... we can define the notion of "inverse cardinality" with the "owners" function

inverse cardinality of a node $$||a||_{\in_\alpha}=|own_{\in_\alpha}(a)|$$

example "$\{V_1\}$" appears as node in all the graphs $(V_\alpha,\in_\alpha)$ for $\alpha>2$ and while $|\{V_1\}|_{\in_\alpha}=1$ holds for every $\alpha>2$ $||\{V_1\}||_{\in_\alpha}$ doesnt. The value grow:

$||\{V_1\}||_{\in_3}=0$, $||\{V_1\}||_{\in_4}=9$

note 3 we now can see that the number of edges of a graph are exatly the sum of the cardinalities or of the inverse cardinalities of the nodes that graph

$$ |\in_\alpha|=\sum_{a \in V_\alpha}|a|_{\in_\alpha}=\sum_{a \in V_\alpha}||a||_{\in_\alpha} $$

but, as Asaf Karagila pointed out, the value the inverse cardinality of the nodes of start to assume a constant (infinite) value when $\alpha$ gets enough big. In the case of ZFC, in fact, we can build infinite sets that have a set $a$ as element.

$Q3.$ Exist a more general concept of "inverse cardinality" in $ZFC$ for normal sets? Does have a special name/properties? There is a formula for it for graphs with finite nodes ($V_{\alpha<\omega}$) (see the example)?

After editing and adding info in this question I can feel that I did some big mistake somewhere... but I don't know where. I hope that who try to answer will point out my misconceptions too.

Bounty added: reference for standard terminology is asked, and at least one of the question Q2 and Q3

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Without being able to fully answer your questions, let me say that I don't know if this idea is quite useful. For two reasons: (1) We usually don't care very much what are the elements of our sets (in fact, we often assume our sets are sets of ordinals, or sets of ordinals, and not whatever they were originally); and (2) pretty much any set can be a member of any other set, in the sense that at some point the "inverse cardinality" will begin to take the same constant values for all the elements of $V_\alpha$. –  Asaf Karagila Apr 25 '14 at 6:42
The inverse cardinal as defined would be $|V|$, the "ultimate cardinality" for every set $a$. There is an almost trivial injection from $V$ into the class $\{x\mid a\in x\}$. –  Asaf Karagila Apr 25 '14 at 14:25
yea! @AsafKaragila you are right... was trivial...but is trivial for the "graph version" a.k.a when the nodes are infinite but form a set $V_\alpha$? –  MphLee Apr 25 '14 at 14:29
I'm not quite sure what you mean in that question. –  Asaf Karagila Apr 25 '14 at 14:32
I was not clear at all, sorry.What I mean is if the question is trivial. For the inverse cardinality of normal set we get a cosntant function. What I ask is: for other graphs (when the nodes form a set) does the inverse cardinality still becomes costant (boring and not interesting)? –  MphLee Apr 25 '14 at 14:37

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