Projective well-orders and coanalytic witnesses

2021/06/29 by Bergfalk, Jeffrey, Fischer, Vera, Switzer, Corey Bacal · 1 citation
#03E17 (primary) #03E35 #03E50 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2106.15359

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 \mathfraka=\mathfraku=\mathfraki=ℵ₁<2^ℵ₀=ℵ₂, each of \mathfraka, \mathfraku, \mathfraki has a Π¹₁ witness and there is a Δ¹₃ 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 Δ¹₃ well-order of the reals is consistent with \mathfrakc=ℵ₂ and each of the following: \mathfraka=\mathfraku

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• Coanalytic families of functions

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