  We study the $\mathbb{CP}^2$-slicing number of knots, i.e. the smallest
$m\geq 0$ such that a knot $K\subseteq S^3$ bounds a properly embedded,
null-homologous disk in a punctured connected sum
$(\#^m\mathbb{CP}^2)^{\times}$. We give a lower bound on the smooth
$\mathbb{CP}^2$-slicing number of a knot in terms of its double branched cover,
and we find knots with arbitrarily large but finite smooth
$\mathbb{CP}^2$-slicing number. We also give an upper bound on the topological
$\mathbb{CP}^2$-slicing number in terms of the Seifert form and find knots for
which the smooth and topological $\mathbb{CP}^2$-slicing numbers are both
finite, nonzero, and distinct.
