Question

Suppose $\lambda$ is a strong limit cardinal of cofinality $\omega$ and for $A$ a transitive set, define $L(A)$ in the usual fashion by setting $$L_0(A)=A;$$ $$L_{\alpha+1}(A) = L_\alpha (A)\cup \mathcal P_{Def}(L_\alpha(A));$$ $$L(A)=\bigcup_{\alpha\in Ord} L_\alpha (A).$$

In Woodin's longer article "The Continuum Hypothesis" (in LNL 19, Logic Colloquium 2000), the following facts are stated regarding $L(V_{\lambda+1})$:

(1) If $c$ is Cohen generic over $V$ then very likely $$(L(V_{\lambda+1}))^{V[c]}\neq L(V_{\lambda+1})[c].$$

(2) On the other hand, if $G\subset Coll(\omega_1,\mathbb{R})$ is $V$-generic then $$(L(V_{\lambda+1}))^{V[G]}= L(V_{\lambda+1})[G].$$

Can anyone give a (sketch of) proof of either (1) or (2)? Are these results given only in the context of a non-trivial elementary embedding $j:L(V_{\lambda+1})\prec L(V_{\lambda+1})$ with $crit(j)<\lambda$?

More generally, for a partial order $\mathbb{P}$ and a $G\subset \mathbb{P}$ which is $V$-generic, which properties of $\mathbb{P}$ are sufficient to ensure the equality $$(L(V_{\lambda+1}))^{V[G]}= L(V_{\lambda+1})[G]$$ holds? Fails? Is this even known?

Answer 1

They are immediate consequences of Theorem 175 of W. Hugh Woodin, "Suitable extender models II: beyond $\omega$-huge", J. Math. Log., vol. 11 (2011), no. 2, pp. 115–436, which says

If there is a (proper) elementary embedding $j: L(V_{\lambda+1}) \to L(V_{\lambda+1})$ with $\text{crit}(j)<\lambda$ and if $G\subset \mathbb{P}$ is $V$-generic for some poset $\mathbb{P}\in V_\lambda$, then $V_{\lambda+1} \in L_\lambda(V_{\lambda+1})^{V[G]}$ if and only if $(\lambda^\omega)^V = (\lambda^\omega)^{V[G]}$.

This theorem can be proved using the Large Perfect Set Theorem for subsets of $V_{\lambda+1}$ in $L_\lambda(V_{\lambda+1})$, which says assuming there is a (proper) elementary embedding $j: L(V_{\lambda+1}) \to L(V_{\lambda+1})$ with $\text{crit}(j)<\lambda$, those subsets of $V_{\lambda+1}$ either has size $\leq \lambda$ or contains a large perfect subset, i.e. a homeomorphic copy of $\lambda^\omega$. Cramer recently improves the Large Perfect Set Theorem to all subsets of $V_{\lambda+1}$ in $L(V_{\lambda+1})$, so $L_\lambda(V_{\lambda+1})$ can be replaced by $L(V_{\lambda+1})$.