Is it consistent with ZC that a well-order of type $\omega_\omega$ does not exist?

Question

Working in Zermelo's set theory (with choice for simplicity) - the construction in Hartogs' theorem shows that starting with a set $X$, there is a set $X'$ in at most $\mathcal{P}^4(X)$ (where $\mathcal P$ denotes power set) and a well-order $<$ in at most $\mathcal{P}^6(X)$ such that the well-order type of $(X',<)$ is the least well-order such that $X'$ is not injectible into $X$.

It is known that $V_{\omega+\omega}$ is model of $ZC$ so a model of $ZC$ may not have the von-Neumann ordinal $\omega+\omega$. However, it does have well order types corresponding to $\omega_n$ for all $n\in\omega$. Start with $\omega$. Construct a set $P_1$ by the Hartogs procedure with well-order type $\omega_1$. Then continue inductively to construct $P_n$ with well-order type $\omega_n$ for all $n\in\omega$.

If we want to continue to construct a well-order of type $\omega_\omega$ - we run into a block. Since $\{P_n\colon n\in\omega\}$ may not be a set - so there's no axiom making it possible to take union or disjoint union of the $P_n$.

However, this does not disprove that a set with well-order type $\omega_\omega$ or cardinality $\aleph_\omega$ exists.

1. Is it consistent with ZC that there is no well-ordered set whose well-order type is greater than the type of $\omega_n$ for all $n\in\omega$?
2. Is it consistent with ZC that there is no set whose cardinality is strictly greater than the cardinality of $P_n$ for all $n\in\omega$?

Answer 1

Yes. Work in a model of $ZFC$ plus the generalized continuum hypothesis and consider the substructure $V_{\omega\cdot 2}\vDash ZC$. Every element of $V_{\omega\cdot 2}=\bigcup_{\alpha<\omega\cdot 2}V_\alpha$ will have cardinality less than some $\beth_n=\aleph_n<\aleph_\omega$. Since you don't have replacement in $V_{\omega\cdot 2}$ you can no longer speak about aleph numbers, however you have that in $V_{\omega\cdot 2}$ any sequence of infinite sets $(P_n)_{n\in\omega}$ will be contained in some $V_{\omega +m}$. If $V_{\omega\cdot 2}\vDash\forall n\in\omega\;|P_{n+1}|>|P_{n}|$ then you would have that $\beth_m=|V_{\omega+m}|>\aleph_\omega$ which contradicts our assumption that the generalized continuum hypothesis holds.

edit: You need to be cautious in general since cardinality is not absolute between transitive structures, however it is for structures of the form $V_\alpha$ for $\alpha$ a limit ordinal.