Author:
MÜLLER SANDRA,SARGSYAN GRIGOR
Abstract
AbstractWe analyze the hereditarily ordinal definable sets
$\operatorname {HOD} $
in
$M_n(x)[g]$
for a Turing cone of reals x, where
$M_n(x)$
is the canonical inner model with n Woodin cardinals build over x and g is generic over
$M_n(x)$
for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming
$\boldsymbol \Pi ^1_{n+2}$
-determinacy, for a Turing cone of reals x,
$\operatorname {HOD} ^{M_n(x)[g]} = M_n(\mathcal {M}_{\infty } | \kappa _{\infty }, \Lambda ),$
where
$\mathcal {M}_{\infty }$
is a direct limit of iterates of
$M_{n+1}$
,
$\delta _{\infty }$
is the least Woodin cardinal in
$\mathcal {M}_{\infty }$
,
$\kappa _{\infty }$
is the least inaccessible cardinal in
$\mathcal {M}_{\infty }$
above
$\delta _{\infty }$
, and
$\Lambda $
is a partial iteration strategy for
$\mathcal {M}_{\infty }$
. It will also be shown that under the same hypothesis
$\operatorname {HOD}^{M_n(x)[g]} $
satisfies
$\operatorname {GCH} $
.
Publisher
Cambridge University Press (CUP)
Reference24 articles.
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3. Fine Structure and Iteration Trees
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