α- Properness and Not Adding Reals
Next to not collapsing ℵ 1 not adding reals seems the most natural requirement on forcing notion. There are many works deducing various assertions from CH and many others who did it from diamond of ℵ 1. If we want to show that the use of diamond is necessary, we usually have to build a model of ZFC in which CH holds but the assertion fails, by iterating suitable forcing. A crucial part in such a proof is showing that the forcing notions do not add reals even when we iterate them.
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