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Mar 29, 2014 at 2:15 comment added Joel David Hamkins I don't know, but I suspect yes. I do know that $\text{MP}(\mathbb{R})$ does not necessarily imply projective absoluteness, because there is a mis-match in the large cardinal strength. Thus, projective truth can be changeable by forcing, even when $\text{MP}(\mathbb{R})$ holds, and this seems close to what you want.
Mar 29, 2014 at 2:09 comment added Noah Schweber Is it known if "$MP(\mathbb{R})+$there are such switching $A$ and $B$" is consistent with $ZFC$?
Mar 29, 2014 at 1:52 history edited Joel David Hamkins CC BY-SA 3.0
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Mar 29, 2014 at 1:39 history answered Joel David Hamkins CC BY-SA 3.0