Call an ordered pair of formulae $\langle P(\kappa,\tilde{a}), Q(\nu,\tilde{a})\rangle$ in the language of $\{\in\}$ *unproblematic* iff

- ZFC proves that for all $\tilde{a}$ and all cardinal numbers $\kappa$ and $\nu$, if $P(\kappa,\tilde{a})$ and $Q(\nu,\tilde{a})$ hold, then $\kappa \leq \nu$ as cardinal numbers.
- ZFC does not prove that for some $\tilde{a},$ there exists a cardinal number $\kappa$ such that $P(\kappa,\tilde{a})$ and $Q(\kappa,\tilde{a})$ hold.

Then clearly, if $\langle P(\kappa,\tilde{a}), Q(\nu,\tilde{a})\rangle$ is an unproblematic formula pair, then we may adjoin the following axiom to ZFC and be confident that the resulting theory will be consistent.

Axiom.For all $\tilde{a}$ and all cardinal numbers $\kappa$ and $\nu$, if $P(\kappa,\tilde{a})$ and $Q(\nu,\tilde{a})$ hold, then $\kappa < \nu$ as cardinal numbers.

**My question is this.** Suppose we have a family of ordered pairs of formulae $$\Phi=\{\langle P_i(\kappa,\tilde{a}), Q_i(\nu,\tilde{a})\rangle\}_{i \in I}$$ such that $\Phi_i$ is unproblematic for every $i \in I$. If we adjoin the corresponding family of axioms to ZFC, is the resulting theory necessarily consistent? And if so, is this extension necessarily conservative over ZFC for sentences in the language of first-order arithmetic?