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Session 32. Set Theory
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Full splitting Miller trees and ioe reals |
Giorgio Laguzzi, Universität Hamburg, Germany
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The talk is based on the joint work with Yurii Khomskii
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We investigate two tree forcings for adding infinitely often equal reals: the full splitting Miller forcing \(\mathbb{FM}\), introduced by Rosłanowski in [1], and the infinitely often equal trees forcing \(\mathbb{IE}\), implicitly introduced by Spinas in [2]. We prove results about Marczewski-type regularity properties associated with these forcings as well as dichotomy properties on \(\mathbf{\Delta}^1_2\) and \(\mathbf{\Sigma}^1_2\) levels, with a particular emphasis on a parallel with the Baire property. Furthermore, we prove that our dichotomies hold for all projective sets in Solovay's model, and that the use of an inaccessible is necessary for both.
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References- A. Rosłanowski, On game ideals , Colloq. Math., 59(2):159-168, 1990.
- O. Spinas, Perfect set theorems , Fund. Math., 201(2):179-195, 2008.
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