The intention behind this posting is to arrive at a definition of cardinality that can work in $\sf ZF$, and at the same time doesn't exhaust the whole of $V$. The known definition uses Scott's trick, so a cardinal is defined as an equivalence class under bijection of sets of the lowest possible rank. The problem is that this definition exhausts the whole of the universe, that is we have: $$\forall x \exists c: c \text { is a Scott cardinal} \land x \in TC(c)$$. Now, if we have two transitive models $M;N$ of $\sf ZF$, and if the set $\operatorname {Card}^M$ of all Scott cardinals in $M$ is equal to $\operatorname {Card}^N$, then we must have $N=M$. This is a shortcoming in the implementation of Cardinality, since it need not exhaust the whole universe with it.

The usual definition of Cardinality in $\sf ZFC$ after Von Neumann's obviousely doesn't have this shorcoming. Scott cardinals will not enable us to speak of distinct transitive models having the same set of cardinals.

What is needed is a restriction on this definition, we need a predicate $P$ that fulfills the following two sentences: $$\forall x \, \exists c: x \sim c \land P(c) \\ \neg \forall x \exists c: P(c) \land x \in TC(c)$$ where $\sim$ stands for "existence of a bijection".

I think we can take $P$ to be *hereditarily_(empty or singleton or a set of empty or singletons)* to qualify for the above, formally: $$P(x) \iff \forall y \in TC(\{x\}): \\\operatorname {y=\emptyset \lor singleton}(y) \lor \\\forall z \in y \, (\operatorname {z=\emptyset \lor singleton}(z))$$

I think that would satisfy the above two requirements in $\sf ZF$.

Now we define a modified Scott cardinal as an equivalence class under "bijection" of $P$-sets of the lowest possible rank.

Are there other known examples that would satisfy the requirements here? Especially ones that have less structure or ones that have simpler definition?