Its known that within the perspective of $\sf ZF$ related theories, Scott's definition of cardinality can work under weaker grounds than Von Neumann's! Scott's cardinality works as long as the principle "*every set is equinumerous to some well founded set*" is in action. Now choice implies that, so it works under choice (even in absence of regularity!), and it can also work in absence of choice and regularity but provided that the above principle is axiomatized.



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> Can we have a definition of cardinality that is weaker than that of Scott's? That is, it works under all grounds that Scott's cardinality works under, but it can also work at some grounds where Scott's cardinality fails!
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[Scott's cardinality](https://en.wikipedia.org/wiki/Scott%27s_trick) of a set $x$ is the set of all sets equinumerous to $x$ of the lowest possible rank.



Can we define cardinality that works under weaker grounds than Scott's cardinals?
The answer is YES!


Let $\mathcal H\_\alpha$ stand for the set of all sets hereditarily strictly subnumerous to ordinal $\alpha$.


Now for any set $x$, $\mathcal H^x\_{min}$ is meant to be the minimal $\mathcal H\_\alpha$ such that there exists an iterative power of it that is supernumerous to $x$. Formally:


Define: $\mathcal H^x\_{min} = min \ \mathcal H\_\alpha: \exists \beta [ x \rightarrowtail P\_\beta(\mathcal H\_\alpha)]$


Where $``\rightarrowtail"$ signify "*is injective to*".


$P\_\beta$ is defined recursively as:


$P\_\emptyset(x)=x \\ P\_{\beta+1}(x)= P(P\_\beta(x)) \\ P\_\beta(x) = \bigcup (\{P\_\alpha(x):\alpha < \beta\}) \text{ if } \beta \text{ is a limit ordinal} $


Now by $P^x\_{min}(S) $ its meant the minimal iterative power of $S$ that is supernumerous to $x$. Formally:


Define: $P^x\_{min} (S) = min \ P\_\beta (S): x \rightarrowtail P\_\beta(S) $


Now we come to define *cardinality* of a set $x$, denoted by $``|x|"$, as the set of all subsets of $P^x\_{min} (\mathcal H^x\_{min})$ , that are equinumerous to $x$. Formally:


Define: $|x|= \{y| \ y \sim x \land y \subseteq P^x\_{min} (\mathcal H^x\_{min}) \}$


Where $\sim$ signify "*is bijective to*"


This definition of cardinality can work under grounds weaker than those of Scott's cardinality? The latter demands the assumption that "*every set is equinumerous to some well founded set*", and under that assumption the cardinality [defined here] of any set would be exactly its Scott's cardinal. But this definition can work even when the above assumption fails, but it requires the statement: $$\forall x \exists \alpha \exists \beta : x \rightarrowtail P\_\beta (\mathcal H\_\alpha)$$ which doesn't imply the above assumption! (See this [answer](https://mathoverflow.net/questions/309562/can-cardinality-be-defined-with-essentially-no-practical-restriction-on-non-well?answertab=active#tab-top), and [this](https://math.stackexchange.com/questions/3740391/can-we-have-a-converse-skolem-paradox)).


Note: by *well founded set* its meant a set whose transitive closure is well founded with respect to $\in$.


