Since \(\mathbf{\Pi}^1_1\)-determinacy is a desirable property on the reals, the natural question arises as to how one can preserve it under forcing. We will show using the technique of capturing that the statement 'Every real has a sharp' is preserved under any countable support iteration of 'simply' definable forcing notions. By the famous results of L. Harrington and D. Martin this shows that \(\mathbf{\Pi}^1_1\)-determinacy is preserved under such iterations.

More generally, our theorem also shows that the statement '\(M_n^\sharp(x)\) exists for every real \(x \in \omega^\omega\)' is preserved. By the results of I. Neeman and H. Woodin this generalizes our result to higher levels of projective determinacy.

Without the existence of large cardinals the technique of capturing can still be used to show preservation results for regularity properties such as the \(\mathbf{\Delta}^1_2\)- or \(\mathbf{\Sigma}^1_2\)-Baire property.

This is a joint project with J. Schilhan and P. Schlicht.