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UNCOUNTABLE TREES AND COHEN $\kappa$-REALS

Published online by Cambridge University Press:  02 July 2019

GIORGIO LAGUZZI
GIORGIO LAGUZZI*
Affiliation:
ALBERT-LUDWIG-UNIVERSITAET-FREIBURG MATHEMATISCHES INSTITUTE ERNST-ZERMELO STR. 1, OFFICE311 79104FREIBURG IM BREISGAU, GERMANY E-mail: giorgio.laguzzi@libero.it
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Abstract

We investigate some versions of amoeba for tree-forcings in the generalized Cantor and Baire spaces. This answers [10, Question 3.20] and generalizes a line of research that in the standard case has been studied in [11], [13], and [7]. Moreover, we also answer questions posed in [3] by Friedman, Khomskii, and Kulikov, about the relationships between regularity properties at uncountable cardinals. We show ${\bf{\Sigma }}_1^1$-counterexamples to some regularity properties related to trees without club splitting. In particular we prove a strong relationship between the Ramsey and the Baire properties, in slight contrast with the standard case.

Keywords

03Exxforcingdescriptive set theory

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 3 , September 2019 , pp. 877 - 894
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

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