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Sep 17 at 16:12 vote accept Zoorado
Sep 17 at 11:07 history became hot network question
Sep 17 at 11:04 answer added Joel David Hamkins timeline score: 16
Sep 17 at 10:57 comment added Miha Habič There is a zoo of forcing notions adding interesting objects (Suslin trees, clubs, $\square$ sequences,...) with "finite conditions", which would satisfy your property. They are often proper, but I'm not sure whether they add reals or not.
Sep 17 at 6:43 history edited Zoorado CC BY-SA 4.0
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Sep 17 at 3:57 history edited Noah Schweber
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Sep 17 at 2:48 history asked Zoorado CC BY-SA 4.0