In this talk I will address the prospect of obtaining arbitrary forcible (over $V$) $\Sigma_2$ sentences by forcing over a fixed model of the form $L(V_\delta)$. This will motivate a ``local'' version of $\Omega$-logic for which one can prove a completeness theorem in the spirit of Woodin's $\Omega$-conjecture. Time permitting, I will present partial results (joint with R.\ Schindler) on the consistency of $MM^{++, *}$, a natural enhancement of $MM^{++}$ involving the strong notion of consistency at play in the definition of $\Omega$-logic.