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AN ANALYSIS OF THE MODELS $L[T_{2n} ]$

Published online by Cambridge University Press:  04 February 2019

RACHID ATMAI
RACHID ATMAI*
Affiliation:
DEPARTMENT OF MATHEMATICS MIRACOSTA COLLEGE OCEANSIDE, CA 92056, USAE-mail: ratmai@miracosta.edu
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Abstract

We analyze the models $L[T_{2n} ]$, where $T_{2n}$ is a tree on $\omega\times \kappa _{2n + 1}^1 $ projecting to a universal ${\rm{\Pi }}_{2n}^1 $ set of reals, for $n > 1$. Following Hjorth’s work on $L[T_2 ]$, we show that under ${\rm{Det}}\left( {{\rm{}}_{2n}^1 } \right)$, the models $L[T_{2n} ]$ are unique, that is they do not depend of the choice of the tree $T_{2n}$. This requires a generalization of the Kechris–Martin theorem to all pointclasses${\rm{\Pi }}_{2n + 1}^1$. We then characterize these models as constructible models relative to the direct limit of all countable nondropping iterates of${\cal M}_{2n + 1}^\#$. We then show that the GCH holds in $L[T_{2n} ]$, for every $n < \omega $, even though they are not extender models. This analysis localizes the HOD analysis of Steel and Woodin at the even levels of the projective hierarchy.

Keywords

03E1503E45descriptive set theoryinner model theory

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Articles
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The Journal of Symbolic Logic , Volume 84 , Issue 1 , March 2019 , pp. 1 - 26
Copyright
Copyright © The Association for Symbolic Logic 2019 

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