Projective well orders and coanalytic witnesses |
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Institution: | Institut für Mathematik, Kurt Gödel Research Center, Universität Wien, Kolingasse 14-16, 1090 Wien, Austria |
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Abstract: | We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable P-points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which , each of , , has a witness and there is a well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-order are optimal. In addition, we show that the existence of a well-order of the reals is consistent with and each of the following: , , , where the smaller cardinal characteristics have co-analytic witnesses.Our methods allow the preservation of only sufficiently definable witnesses, which significantly differs from other preservation results of this type. |
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Keywords: | Definable well-orders of the reals Projective combinatorial sets of reals Cardinal characteristics Forcing preservation theorems |
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