# Can there be a minimal remote cardinal?

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: $$\exists x : \operatorname {remote}(x) \land \forall y \, ( \operatorname {remote}(y) \to |x| \leq |y|)$$

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. $$\square$$