Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The complexity of the reals in inner models of set theory   The usual definition of the set of constructible reals is $\Sigma ^1_2$. This
set can have a simpler definition if, for example, it is countable or if every
real is constructible.
  H. Friedman asked if the set of constructible reals can be analytic or even
Borel in a nontrivial way. A related problem was posed by K. Prikry: can there
exist a nonconstructible perfect set of constructible reals? The main result of
this paper is a negative answer to Friedman's question. In fact we prove that
if $M$ is an inner model of set theory and the set of reals in $M$ is analytic
then either all reals are in $M$ or else $\aleph _1^M$ is countable. We also
extend this result to higher levels of the projective hierarchy under
appropriate large cardinal assumptions.
  Concerning Prikry's problem we show that the answer is negative if "perfect"
is replaced by "superperfect" but that it can be positive if "constructible" is
replaced by "belonging to some inner model $M$".

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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: solv-int
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The categories of this paper are: eess
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: adap-org
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The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th