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Inner mantles and iterated HOD
Author(s) -
Reitz Jonas,
Williams Kameryn J.
Publication year - 2019
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201800071
Subject(s) - iterated function , conjecture , mathematics , extension (predicate logic) , sequence (biology) , class (philosophy) , forcing (mathematics) , pure mathematics , combinatorics , mathematical analysis , chemistry , computer science , artificial intelligence , biochemistry , programming language
We present a class forcing notion M ( η ) , uniformly definable for ordinals η, which forces the ground model to be the ηth inner mantle of the extension, in which the sequence of inner mantles has length at least η. This answers a conjecture of Fuchs, Hamkins, and Reitz [1] in the positive. We also show that M ( η ) forces the ground model to be the ηth iterated HOD of the extension, where the sequence of iterated HOD s has length at least η. We conclude by showing that the lengths of the sequences of inner mantles and of iterated HOD s can be separated to be any two ordinals you please.