5 problems

Assume the hypotheses and notation of the paper's comparison theorem: $\mathcal{M}$ and $\mathcal{N}$ are countable premice, with the specified iterability, smallness, realizabilit…

Assume $\mathrm{ZF}+\mathsf{AD}^{L(\mathbb R)}$. Let $M$ be a $(0,\omega_1)$-iterable countable $\omega$-small premouse satisfying “$\omega_1$ exists”. Then there are $\gamma$,…

Assume $\mathsf{AD}^{L(\mathbb R)}$. Let $M$ be an iterable countable $\omega$-small premouse. Then there are $\gamma\in\mathrm{Lim}_0$, $\beta\in\mathrm{Lim}_0\cup\{\mathrm{OR}\}$…

Let $M$ be a countable, $\omega$-small, $(0,\omega_1+1)$-iterable premouse. Steel's conjecture. There is a wellorder of $\mathbb R\cap M$ in $L(\mathbb R\cap M)$. This is Steel's s…

Let $m<\omega$ and let $M$ be an $m$-sound, $(m,\omega_1+1)$-iterable premouse, and let $M$ be active, $1$-sound, and $(0,\omega_1+1)$-iterable in the second case. The questions ar…