Uniform unfolding and analytic measurability

نویسنده

  • Benedikt Löwe
چکیده

There are many natural forcing notions adding real numbers, most notably the precursors of this class of forcing notions, namely Cohen and Random forcing. Whereas Random forcing has been deliberately constructed in connection with the σ–algebra of Lebesgue measurable sets and Cohen forcing is represented by the algebra of Borel sets modulo the σ–ideal of meager sets, the other classical forcing notions, which add real numbers, e.g. Sacks forcing, Miller forcing and Laver forcing are not connected with a σ–algebra by virtue of their construction. But as one can see easily, each of these classical forcing notions corresponds naturally to a σ–algebra. Miller alludes to this fact in the rather vague formulated problem (11.10) of his problem list. The main point of the problem is: What kinds of well–known results can be proved for these σ–algebras? E.g. it is known that the analytic sets are Lebesgue measurable and have the Baire property, and we have the first sets which do not have this property on the ∆2 level in L. So, given a forcing notion P adding real numbers, the following two natural questions are connected with Miller’s problem (11.10):

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عنوان ژورنال:
  • Arch. Math. Log.

دوره 37  شماره 

صفحات  -

تاریخ انتشار 1998