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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?

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Some context missing as there is no free access to that paper, for example what is $\lambda$, and Woodin's definitions of () and []. If V[c] is a Cohen extension and $\lambda$ small enough then $(L(V_{\lambda+1}))^{V[c]} = L(V_{\lambda+1})^{V}$. – Eran Oct 12 '12 at 9:41
Eran: I added the context you requested. Would you mind elaborating on your remark about Cohen extensions and small $\lambda$? – Everett Piper Oct 12 '12 at 16:38
Eran: If (as I assumed) "Cohen generic" refers to a Cohen real (as opposed to a Cohen subset of some larger ordinal), then $(L(V_{\lambda+1}))^{V[c]}$ will differ from $L(V_{\lambda+1})^V$ as soon as $\lambda\geq\omega$, because the former contains $c$ and the latter doesn't. – Andreas Blass Oct 12 '12 at 16:53
Andreas, I think that Everett may be intending it to be read as $L(V_{\lambda+1}^{V[c]})$, that is, as the "L(V_{\lambda+1})" of the model $V[c]$. – Joel David Hamkins Oct 12 '12 at 17:00
If you allow $c$ (instead of Cohen generic real) to be a real coding two Sacks generic reals, then as in (…) $(L(V_{\lambda+1}))^{V[c]}$ will contain their minimal degrees while $L(V_{\lambda+1})[c]$ won't. – Eran Oct 13 '12 at 1:52
up vote 8 down vote accepted

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 $L(V_{\lambda+1})$.

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X. Shi: Thank you so much for this answer. I have been immersed in the Large Perfect Set Theorem for a week now. I have two related questions as a result. (1) In your paper on the Robinson-Posner Theorem at $I_0$, you state the Large Perfect Set Theorem for $X\subset V_{\lambda+1}$ definable from parameters in $V_{\lambda+1}$, i.e., for $X\in L_1(V_{\lambda+1})$. Can you extend your argument there in some way to get the result for all large $X\in L_\lambda(V_{\lambda+1})$? I ask because Woodin's proof of Large Perfect Set in Lemma 22 uses the notion of $\mathbb{U}(j)$-representability. – Everett Piper Oct 19 '12 at 23:14
(2) Can you sketch Cramer's extension of Large Perfect Set Theorem to the more general case where $X\in L(V_{\lambda+1})$? – Everett Piper Oct 19 '12 at 23:15

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