# Are there some interesting propositions independent with ZF+V=L that do not increase consistency strength?

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)?

• 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. Sep 17 at 6:53
• @CoreyBacalSwitzer Your comment deserves to be posted as an answer IMO. Sep 18 at 15:31