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Working in $\sf Z- Reg.$ we can have sets bigger than every well-founded set, let's label such sets as "remote". Now, suppose we'll add the axiom of existence of transitive closures, and the axiom stating that for every set $x$ there exists a set $\mathcal H_x$ of all sets hereditarily strictly subnumerous to $x$ and such that there exists an injection from $x$ to $\mathcal H_x$. This is compatible with the existence of remote sets. The point is that we can work in a milieu outside of regularity and choice where the notion of cardinality is definable! So, here we define a cardinal as an equivalence class of all sets hereditarily strictly subnumerous to a common set, under equivalence relation bijection. So the cardinality of $x$, denoted by "$|x|$", is the set of all ranges of injections from $x$ to $\mathcal H_x$. Now, the cardinalities of remote sets shall be called "remote cardinals".

Now, is it consistent to add an axiom stating the existence of a set of minimal remote cardinality? That is: $$ \exists x : \operatorname {remote}(x) \land \forall y \, ( \operatorname {remote}(y) \to |x| \leq |y|)$$

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Working in "$\sf ZFC$-$\sf Reg. $ + for every set there is a set of Quine atoms as big as it", lets take a set $S$ whose cardinality is the minimal cardinal strictly bigger than the cardinality of any set in $V_{\omega+\omega}$. Now let's take a set $A$ of Quine atoms, of the same cardinality as $S$. Let $V_{\omega+\omega}(A)$ be the set obtained from iteratively powering $A$ along $\omega+\omega$ in the usual manner. Whenever externally/internally is mentioned from hereafter it is with regards to outside and inside of $V_{\omega+\omega}(A)$ in the context of speaking of $\langle V_{\omega+\omega}(A), \in \restriction_{V_{\omega+\omega}(A)}\rangle $ as a model of $\sf Z$-$\sf Reg$. Now, externally speaking every element of $V_{\omega+\omega}$ is strictly smaller than $A$, but those are the ones seen inside $V_{\omega+\omega}(A)$ as well-founded, and so none of them can be seen from the inside to be bigger or equal in cardinality to $A$, since $V_{\omega+\omega}(A)$ is supertransitive then all injections from these sets to $A$ are preserved internally, then all the internally seen well-founded sets would be also internally seen to be strictly subnumerous to $A$. Since, $A$ is externally of minimal cardinality above all cardinalities of elements of $V_{\omega+\omega}$, then it would retain that status internally. If otherwise, then this mean that there is a set $B$ that is internally looking as remote but externally it is not so, that is $B$ externally has an injection to some element of $V_{\omega+\omega}$ yet that injection is not seen internally (i.e. not an element of $V_{\omega+\omega}(A)$), contradicting supertransitivity of $V_{\omega+\omega}(A)$, also for a proof by negation if we hold the existence of such a set $B$, then there will exist an element $C$ of $V_{\omega+\omega}$ that is externally strictly supernumerous to $B$ [for example $C=\mathcal P(B)$], and so $C$ would be seen as incomparable to $B$ internally, and so $B$ is not internally seen as remote. $\square$

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