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Let $\kappa$ be an inaccessible cardinal. Is there any forcing notion $P$ with the following properties: 1-$P$ preserves GCH and the strong inaccessibility of $\kappa$, 2-$P$ adds a subset of $\kappa$ of size $\kappa$, 3-conditions in 3-$P$ is the $< \aleph_1-$support product of some forcing notions $P_{\alpha}, \alpha < \kappa.$ 4- The generic filter for $P$ are at most countablecan be reconstructed from the subset of $\kappa$ added in 2. |
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Adding large sets by countable conditions preserving the GCHLet $\kappa$ be an inaccessible cardinal. Is there any forcing notion $P$ with the following properties: 1-$P$ preserves GCH and the strong inaccessibility of $\kappa$, 2-$P$ adds a subset of $\kappa$ of size $\kappa$, 3-conditions in $P$ are at most countable.
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