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Let $SF$ be the schema of stratified comprehension.

Take the theory $SF + Infinity + Choice + \text {Extensionality fails everywhere}$.

Are the following consistent with this theory?

  1. $\forall X (|X| \leq |P_1(X)|)$

  2. $\forall X (Infinite(X) \to |X|=|P_1(X)|)$

It is known that these two statements fail in NFU, since NFU proves $|P_1(X)| < |X|$ for some sets, which makes it guilty of committing cardinality error of the first kind.

However the known proof of that error in NFU is Extensionality dependent, hence the question. It might be the case that removal of Extensionality may result in avoidance of first kind cardinality error.

It is known that this theory can interpret $NFU + Infinity + Choice$ because SF can interpret NFU a proof due to Marcel Crabbé. An equivalent proof of NFU being interpretable form SF is present here 15-17.

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This is a partial answer, it only answers the first question.

Working in NFU + Infinity + Choice + $|U|>|P(V)|$ [which is consistent relative to ZFC]

Since we have choice then there is a function $H$ from a partition $K$ on $U$ that has all of its pieces (i.e. elements of $K$) equinumerous to $|V|$, to $P1(P1(V))$, i.e. to the set of all double singletons.

Now we form a surjective function $J$ from $V$ to $P(V)$ such that every set is sent by $J$ to itself, while every Ur-element (an empty object other than the empty set) is sent by $J$ to a singleton set in such a manner that each Ur-element $x$ is sent to a singleton $y$ by $J$ (i.e. $J(x)=y$) if and only if there exists a piece $p$ of $K$ such that $x \in p \land y \in H(p)$.

Now we define a new membership relation $E$ as follows:

$y \ E \ x \iff y \in J(x)$

Now the structure $\langle V, E \rangle$ would satisfy all axioms of SF. But Extensionality would be violated over all and only singleton sets in such a manner that each singleton set has $|V|$ many copies.

This way every set in that structure would be subnumerous to the set of all singletons of its elements.

IF instead of $H$ sending $K$ to $P1(P1(V))$ we let it send $K$ to $P1(V)$, and run the same argument, the result is that we'll get a stratified theory in which every object in this theory has $|V|$ many co-extensional copies!

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