Incomparable V γ $V_\gamma$ -degrees

In [3], Shi proved that there exist incomparable Zermelo degrees at γ if there exists an ω-sequence of measurable cardinals, whose limit is γ. He asked whether there is a size ${\gamma }^{\omega }$ antichain of Zermelo degrees. We consider this question for the $V_{\gamma }$-degree structure. We use a kind of Prikry-type forcing to show that if there is an ω-sequence of measurable cardinals, then there are ${\gamma }^{\omega }$-many pairwise incomparable $V_{\gamma }$-degrees, where γ is the limit of the ω-sequence of measurable cardinals.