I need to know if the following system is consistent, because I want to use it in presenting automorphisms over stratified versions of it.


The system I'd label as "*Acyclic ZF*", which is $\small \sf ZF-Reg.+ acyclic \ AFA + Rank$, is obtained by adding a two place predicate symbol $\mathcal R$ symbolizing *is the rank of* to the language of $\small \sf ZF$, then replace Regularity by the following two acyclic AFA axioms:


**Acyclicity:** $\forall x\_1,..,x\_n: \neg (x\_1 \in x\_2 \in x\_3 \in ... \in x\_n \land x\_1=x\_n)$


**Acyclic construction:** For every *acyclic* [accessible pointed graph](https://en.wikipedia.org/wiki/Rooted_graph#Set_theory) there exists a set whose membership graph is isomorphic to it, where the latter means the membership map on the transitive closure of that set.


Now we define the unary predicate *ordinal*, symbolized by $\mathcal Ord$, as *transitive set of transitive sets*. To be emphasized here is that an ordinal can be a von Neumann or may not be so! If the ordinal is well founded on $\in$, then it is a von Neumann ordinal, if not then its to be called as a non-standard ordinal, or even more outrageously an *ill-founded ordinal*. We make axioms to the effect that the ranking relation $\mathcal R$ constitute a partial function from ordinals to sets such that the indexed sets would correspond to iterative powers similar to the buildup of the cumulative hierarchy. Formally this is:


$\forall a,b,c,d: \mathcal R(a,b) \land \mathcal R(c,d) \longrightarrow [a=c \Leftrightarrow b=d]$


$\forall x: \exists y (\mathcal R (x,y)) \iff \mathcal Ord(x)$


$\forall \alpha \forall x \forall y: \mathcal R(\alpha, x) \land \mathcal R (\alpha \cup \{\alpha \}, y) \longrightarrow y=\mathcal P(x)$


$\forall \alpha \forall x [(\not \exists \beta: \alpha=\beta \cup \{\beta\}) \land \mathcal R(\alpha,x) \longrightarrow \\ x= \bigcup \{y: \exists \beta \in \alpha ( \mathcal R(\beta,y)) \}]$


$\forall \alpha \forall x : \mathcal R(\alpha,x) \to \alpha \subseteq x \land \alpha \not \in x$


$\forall \alpha \forall x: \mathcal R (\alpha, x) \to \forall y \in x (y \subseteq x)$


The last axiom is to restrict sets to those itrative stages, I'll consider it as a parallel to foundation, that is:


**Para-foundation:** $\forall x \exists \alpha \exists v : \mathcal R(\alpha, v) \land x \in v$



> 
> Now, is Acyclic ZF consistent?
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Is acyclic ZF consistent?
Acyclic ZF is inconsistent. Let D be the directed graph whose vertices are the natural numbers where (a,b) is an edge if a=0 and a<b, or b>0 and a>b. By Acyclic construction, there is set {x0.x1,x2,...} such
that xi∈j iff (i,j) is in D. Each xi is a transitive set and thus each xi is an ordinal.


(1)Suppose R(xi,yi) for all natural numbers i. Suppose j>0. Then every von Neumann ordinal is in yj.  

Proof: Suppose it is not true and let b be the least von Neumann ordinal which is not in yi for all i>0. We note that for i>0, xi=x(i+1)U{x(i+1)}. By our assumption, for every i>0, b⊆y(i+1). Therefore b∈yi.


Therefore there is a set A of all von Neumann ordinals. But then A is a von Neumann ordinal and thus A∈A.


