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Given a cardinal $\kappa,$ recall that $X \subset \kappa$ is called fresh (over $V$), if $X \notin V,$ but $X \cap \alpha \in V$ for all $\alpha < \kappa.$

Question. Is it consistent that there exists a generic extension $V[G]$ of $V$ (for some suitable $V$), such that:

(A) $V$ and $V[G]$ have the same cardinals and cofinalities,

(B) $V$ and $V[G]$ have the same bounded subsets of $\aleph_\omega,$

(C) There exists a fresh subset $X \in V[G]$ of $\aleph_\omega$ (over $V$), such that $V[G]=V[X],$ and $V[G]$ is a minimal generic extension of $V?$

Remark. See A minimal Prikry-type forcing for singularizing a measurable cardinal and Prikry-type forcing and minimal $α$-degree, where some similar results for a large singular cardinal are proved.

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  • $\begingroup$ (+1) What is the difficulty of reformulating the same methods that work for large singular cardinals in the case of smaller singulars like $\aleph_{\omega}$? $\endgroup$
    – user82740
    Commented Nov 30, 2015 at 5:57

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