In 1985 Woodin showed that if there is a proper class of measurable Woodin cardinals and $V^{B_1}$ and $V^{B_2}$ are generic extensions of $V$ satisfying $\CH$ then $V^{B_1}$ and $V^{B_2}$ agree on all $\Sigma^2_1$-statements. In modern terms this can be reformulated by saying that under the above large cardinal assumption $\ZFC+\CH$ is $\Omega$-complete for $\Sigma^2_1$. Moreover, $\CH$ is the unique $\Sigma^2_1$-statement with this feature in the sense that any other $\Sigma^2_1$-statement with this feature is $\Omega$-equivalent to $\CH$. It is natural to look for other strengthenings of $\ZFC$ that enjoy a greater degree of $\Omega$-completeness. For example, one can ask for recursively enumerable axioms $\A$ such that relative to large cardinal axioms $\ZFC+\A$ is $\Omega$-complete for all of third-order arithmetic. Going further, for each specifiable segment $V_\lambda$ of the universe of sets (for example, one might take $\lambda$ to be the least huge cardinal), one can ask for recursively enumerable axioms $\A$ such that $\ZFC+\A$ is $\Omega$-complete for the theory of $V_\lambda$, relative to large cardinal axioms. If such theories exist, extend one another, and are unique in the sense that any other theory $\A'$ with the same level of $\Omega$-completeness as $A$ is actually $\Omega$-equivalent to $\A$, then this would make for a very strong case for new axioms that settle the theory of $V$ in $\Omega$-logic. In this talk I will motivate the above scenario and sketch a proof that uniqueness must fail. This is joint work with Hugh Woodin. In particular, we show that if there is one such theory that $\Omega$-implies $\CH$ then there is another that $\Omega$-implies $ eg\CH$.