  We define the stabilizing number $\operatorname{sn}(K)$ of a knot $K \subset
S^3$ as the minimal number $n$ of $S^2 \times S^2$ connected summands required
for $K$ to bound a nullhomotopic locally flat disc in $D^4 \# n S^2 \times
S^2$. This quantity is defined when the Arf invariant of $K$ is zero. We show
that $\operatorname{sn}(K)$ is bounded below by signatures and Casson-Gordon
invariants and bounded above by the topological $4$-genus
$g_4^{\operatorname{top}}(K)$. We provide an infinite family of examples with
$\operatorname{sn}(K)<g_4^{\operatorname{top}}(K)$.
