Let $ T$ be a theory written in Mono-sorted first order logic with equality, with extralogical primitives: $<, \in$.

**Define:** $x \leq y \iff x < y \lor x=y$

*Axioms:*

$\textbf{Well ordering: }\\\textit{Transitive:} \ x < y \land y < z \to x < z \\ \textit{Connective:} \ x \neq y \leftrightarrow (x < y \lor y < x) \\ \textit{Well founded:} \ \exists n \in x \to \exists n \in x \forall m \in x (n \leq m)$

$\textbf{Finiteness: }\exists n \in x \to \exists n \in x \forall m \in x (m \leq n)$

$\textbf{Sets: } \forall n \exists! x \forall m (m \in x \leftrightarrow m \leq n \land \phi)$, if $x$ is not free in formula $\phi$.

$\textbf{Closure: } x \in y \to x < y$

$\textbf{Emergence: }\max ( x \setminus y) < \max (y \setminus x) \to x < y $

Where, $\max x$ is the maximal element of $x$ with respect to relation $<$; and $x \setminus y = \{m \in x \mid m \not \in y\}$.

Theory $T$ is an example of an interwoven theme of order and set theory. It is synonymous with $\sf PA$. [see here]

If we remove Finiteness, and add the existence of a limit number, that is:

- $\textbf{Infinity: } \exists l \neq \emptyset: \forall x < l \exists y: x < y < l$

, call the resultant theory $``T - {\sf Fin + I}" $.

What would be the set theory that is synonymous to $T - {\sf Fin + I}$?

My guess is that it would be $\sf Z + Ranks$ plus adding a bijection $j$ to the language of $\sf Z$ from sets to von Neumann ordinals in a manner such that sets are assigned higher ordinals than their elements. The Ranks axiom is $(\forall \alpha \exists V_\alpha)$ where $V_\alpha = \{x \in j^{-1}(\beta) \mid \beta < \alpha \}$. So, the theory would prove Global Choice, Transitive closures, Ranks, etc...

[EDIT:] The above theory had been proved inconsistent as shown in the accepted answer. A possible salvage along the original intentions is to restrict axiom of Emergence to apply to natural numbers (i.e. objects prior to the first infinite number). We call $T$ with this restriction as $T^r$. We add the following axiom:

- $\textbf {Respective: } \forall n \in x (m > n) \land m \in y \to x < y$.

So, the question is re-stated in connection with theory $T^r - {\sf Fin + I + Respective}$.