I am reading Jörg Brendle's Bogota note, and the author claimed that Sacks forcing with countable support product does not satisfy a product lemma. Especially, he mentioned that if $\mathbb{S}_I$, $|I|$-many countable support product of Sacks forcing $\mathbb{S}$, adds $|I|$ Sacks reals $\langle s_i\mid i\in I\rangle$ and $i\neq j$, then $s_j$ is not $\mathbb{S}$-generic over $V[s_i]$.
Let me take $I=2$ for simplicity. One subtle point of the claim is that $\mathbb{S}$ need not be absolute between $V$ and $V[s_0]$. In fact, the product lemma for general forcing shows that $s_1$ would be $\mathbb{S}^V$-generic over $V[s_0]$. Hence I think the author intends to claim that $s_1$ is not $\mathbb{S}^{V[s_0]}$-generic over $V[s_0]$.
However, $\mathbb{S}^V\neq\mathbb{S}^{V[s_0]}$ does not automatically mean $s_1$ is not $\mathbb{S}^{V[s_0]}$-generic over $V[s_0]$. My question is:
How to prove $s_1$ is not $\mathbb{S}^{V[s_0]}$-generic over $V[s_0]$?
I am new at Sacks forcing, so I have no idea how to start. I would appreciate your help!
from Hot Weekly Questions - Mathematics Stack Exchange
Hanul Jeon
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