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textsf {AD}^{+}$ implies $ \\omega _{1}$ is a club $ \\Theta $ -Berkeley cardinal
Following [1], given cardinals
$\\kappa <\\lambda $
, we say
$\\kappa $
is a club
$\\lambda $
-Berkeley cardinal if for every transitive set N of size
$<\\lambda $
such that
$\\kappa \\subseteq N$
, there is a club
$C\\subseteq \\kappa $
with the property that for every
$\\eta \\in C$
, there is an elementary embedding
$j: N\\rightarrow N$
with
$\\mathrm {crit }(j)=\\eta $
. We say
$\\kappa $
is
$\\nu $
-club
$\\lambda $
-Berkeley if
$C\\subseteq \\kappa $
as above is a
$\\nu $
-club. We say
$\\kappa $
is
$\\lambda $
-Berkeley if C is unbounded in
$\\kappa $
. We show that under
$\\textsf {AD}^{+}$
, (1) every regular Suslin cardinal is
$\\omega $
-club
$\\Theta $
-Berkeley (see Theorem 7.1), (2)
$\\omega _1$
is club
$\\Theta $
-Berkeley (see Theorem 3.1 and Theorem 7.1), and (3) the ’s are
$\\Theta $
-Berkeley – in particular,
$\\omega _2$
is
$\\Theta $
-Berkeley (see Remark 7.5). Along the way, we represent regular Suslin cardinals in direct limits as cutpoint cardinals (see Theorem 5.1). This topic has been studied in [31] and [4], albeit from a different point of view. We also show that, assuming
$V=L({\\mathbb {R}})+{\\textsf {AD}}$
,
$\\omega _1$
is not
$\\Theta ^+$
-Berkeley, so the result stated in the title is optimal (see Theorem 9.14 and Theorem 9.19).
Journal Article
mathbf {\\Sigma }_1$ -definability at higher cardinals: Thin sets, almost disjoint families and long well-orders
Given an uncountable cardinal
$\\kappa $
, we consider the question of whether subsets of the power set of
$\\kappa $
that are usually constructed with the help of the axiom of choice are definable by
$\\Sigma _1$
-formulas that only use the cardinal
$\\kappa $
and sets of hereditary cardinality less than
$\\kappa $
as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets definable in this way and use it to generalize two classical nondefinability results to higher cardinals. First, we show that a classical result of Mathias on the complexity of maximal almost disjoint families of sets of natural numbers can be generalized to measurable limits of measurables. Second, we prove that for a limit of countably many measurable cardinals, the existence of a simply definable well-ordering of subsets of
$\\kappa $
of length at least
$\\kappa ^+$
implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the nonexistence of
$\\Sigma _1$
-definitions of certain objects at singular strong limit cardinals. Finally, we show that both large cardinal assumptions and forcing axioms cause analogs of these statements to hold at the first uncountable cardinal
$\\omega _1$
.
Journal Article
Decidability of the class of all the rings $\\mathbb {Z}/m\\mathbb {Z}$ : A problem of Ax
We prove that the class of all the rings
$\\mathbb {Z}/m\\mathbb {Z}$
for all
$m>1$
is decidable. This gives a positive solution to a problem of Ax asked in his celebrated 1968 paper on the elementary theory of finite fields [1, Problem 5, p. 270]. In our proof, we reduce the problem to the decidability of the ring of adeles
$\\mathbb {A}_{\\mathbb {Q}}$
of
$\\mathbb {Q}$
.
Journal Article
Souslin quasi-orders and bi-embeddability of uncountable structures
We provide analogues of the results from Friedman and Motto Ros (2011) and Camerlo, Marcone, and Motto Ros (2013) (which correspond
to the case
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