Let me record an observation that emerged in the comments.

Theorem. There is a model of ZCA in which the GCH holds for all pure sets, but not generally for all sets.

Proof. We work in ZFC, and assume GCH holds up to $\beth_\omega$, but fails at $\beth_{\omega+1}$. Using the methods of my paper

• Joel David Hamkins and Bokai Yao, Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal, arxiv:2204.09766

we can interpret a model of ZFCA as $V[[A]]$, in which $A$ is a set of urelements of size $\beth_{\omega+1}$. Now, chop this model off at rank $\omega+\omega$ to form $W=V_{\omega+\omega}[[A]]$. It is easy to see that this is a model of ZCA. In particular, we have the axiom of choice here, and every set has a well order.

The pure sets of $W$ are exactly the sets in $V_{\omega+\omega}$. And since the GCH holds up to $\beth_\omega$ in the original model $V$, we have the GCH holding in $V_{\omega+\omega}$. So the GCH holds in the pure sets of $W$.

But the GCH fails between $A$ and $P(A)$, since in $V[[A]]$ there are failures of GCH at $\beth_{\omega+1}$. So this is a model of ZCA where the GCH holds in the inner model of pure sets, but fails generally. $\Box$

Since this method proceeds from a ZFC model with a failure of GCH higher up, it doesn't really explain how one might have used urelements to find a violation of GCH directly. But still, I find the situation identified in the theorem interesting and worthwhile to explore.

Let me record an observation that emerged in the comments.

Theorem. There is a model of ZCA in which the GCH holds for all pure sets, but not generally for all sets.

Proof. We work in ZFC, and assume GCH holds up to $\beth_\omega$, but fails at $\beth_{\omega+1}$. Using the methods of my paper

• Joel David Hamkins and Bokai Yao, Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal, arxiv:2204.09766

we can interpret a model of ZFCA as $V[[A]]$, in which $A$ is a set of urelements of size $\beth_{\omega+1}$. Now, chop this model off at rank $\omega+\omega$ to form $W=V_{\omega+\omega}[[A]]$. It is easy to see that this is a model of ZCA. In particular, we have the axiom of choice here, and every set has a well order.

The pure sets of $W$ are exactly the sets in $V_{\omega+\omega}$. And since the GCH holds up to $\beth_\omega$ in the original model $V$, we have the GCH holding in $V_{\omega+\omega}$. So the GCH holds in the pure sets of $W$.

But the GCH fails between $A$ and $P(A)$, since in $V[[A]]$ there are failures of GCH at $\beth_{\omega+1}$. So this is a model of ZCA where the GCH holds in the inner model of pure sets, but fails generally. $\Box$

Since this method proceeds from a ZFC model with a failure of GCH higher up, it doesn't really explain how one might have used urelements to find a violation of GCH directly, which to my understanding might be part of the motivation of the question. But still, I find the situation identified in the theorem interesting and worthwhile to explore. The proof here shows a general method of transferring behavior from high rank in a ZFC model to a occur in a model of ZCA, which does not have that behavior in the pure sets.

Let me record an observation that emerged in the comments.

Theorem. There is a model of ZCA in which the GCH holds for all pure sets, but not generally for all sets.

Proof. We work in ZFC. Using the methods of my paper

• Joel David Hamkins and Bokai Yao, Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal, arxiv:2204.09766

we can interpret a model of ZFCA as $V[[A]]$, in which $A$ is a set of urelements of size $\beth_{\omega+1}$. Now, chop this model off at rank $\omega+\omega$ to form $W=V_{\omega+\omega}[[A]]$. It is easy to see that this is a model of ZCA. In particular, we have the axiom of choice here, and every set has a well order.

The pure sets of $W$ are exactly the sets in $V_{\omega+\omega}$. And since the GCH holds up to $\beth_\omega$ in the original model $V$, we have the GCH holding in $V_{\omega+\omega}$. So the GCH holds in the pure sets of $W$.

But the GCH fails between $A$ and $P(A)$, since in $V[[A]]$ there are failures of GCH at $\beth_{\omega+1}$. So this is a model of ZCA where the GCH holds in the inner model of pure sets, but fails generally. $\Box$

Let me record an observation that emerged in the comments.

Theorem. There is a model of ZCA in which the GCH holds for all pure sets, but not generally for all sets.

Proof. We work in ZFC, and assume GCH holds up to $\beth_\omega$, but fails at $\beth_{\omega+1}$. Using the methods of my paper

• Joel David Hamkins and Bokai Yao, Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal, arxiv:2204.09766

we can interpret a model of ZFCA as $V[[A]]$, in which $A$ is a set of urelements of size $\beth_{\omega+1}$. Now, chop this model off at rank $\omega+\omega$ to form $W=V_{\omega+\omega}[[A]]$. It is easy to see that this is a model of ZCA. In particular, we have the axiom of choice here, and every set has a well order.

The pure sets of $W$ are exactly the sets in $V_{\omega+\omega}$. And since the GCH holds up to $\beth_\omega$ in the original model $V$, we have the GCH holding in $V_{\omega+\omega}$. So the GCH holds in the pure sets of $W$.

But the GCH fails between $A$ and $P(A)$, since in $V[[A]]$ there are failures of GCH at $\beth_{\omega+1}$. So this is a model of ZCA where the GCH holds in the inner model of pure sets, but fails generally. $\Box$

Since this method proceeds from a ZFC model with a failure of GCH higher up, it doesn't really explain how one might have used urelements to find a violation of GCH directly. But still, I find the situation identified in the theorem interesting and worthwhile to explore.