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In some MO questions such as this and this, Hamkins gave some examples that is independent with ZF+V=L, however, all of them increase the consistency strength.

Are there some propositions P, which is interesting in some field of mathematics, and is independent with ZF+V=L, and con(ZFC) proves con(ZF+V=L+P) and con(ZF+V=L+¬P)?

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    $\begingroup$ This is essentially unknown. The only real way we know to produce independence results without increasing consistency is by forcing, which emphatically doesn't allow one to preserve V=L. This is discussed in some detail in Shelah's Logical dreams here: arxiv.org/pdf/math/0211398.pdf, see 4.8 Dream. $\endgroup$ Sep 17 at 6:53
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    $\begingroup$ @CoreyBacalSwitzer Your comment deserves to be posted as an answer IMO. $\endgroup$ Sep 18 at 15:31
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As per @TimothyChow 's suggestion, I'm posting my comment as an answer.

This is essentially unknown. The only real way we know to produce independence results without increasing consistency is by forcing, which emphatically doesn't allow one to preserve V=L. This is discussed in some detail in Shelah's Logical dreams, see https://arxiv.org/pdf/math/0211398.pdf, 4.8 Dream.

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