The class doesn't satisfy a lot of ZFC, it can't even prove that $|X|<2^{|X|}$.

Note that $HPD(\omega)$.

Externally we know that $(2^{|\omega|})^{HPD}$ is countable 

I claim that $HPD$ also see that, this is because $V_{\omega+2}$ has a canonical well-ordering for $(\mathcal P(\omega))^{HPD}$ of ordertype $\omega$ (it is even without parameters), simply by intertwining Ackermann coding for parameters with Godel encoding for formulaes (and noting that every $HPD$ subset of $\omega$ must be seen from $V_{\omega+1}$), so $HPD\models |\omega|=|\mathcal P(\omega)|$