Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Elementary Theories of Ordinal Notation Systems based on Reflection
  Principles   We consider the constructive ordinal notation system for the ordinal
${\epsilon_0}$ that were introduced by L.D. Beklemishev. There are fragments of
this system that are ordinal notation systems for the smaller ordinals
${\omega_n}$ (towers of ${\omega}$-exponentiations of the height $n$). This
systems are based on Japaridze's provability logic $\mathbf{GLP}$. They are
closely related with the technique of ordinal analysis of $\mathbf{PA}$ and
fragments of $\mathbf{PA}$ based on iterated reflection principles. We consider
this notation system and it's fragments as structures with the signatures
selected in a natural way. We prove that the full notation system and it's
fragments, for ordinals ${\ge\omega_4}$, have undecidable elementary theories.
We also prove that the fragments of the full system, for ordinals
${\le\omega_3}$, have decidable elementary theories. We obtain some results
about decidability of elementary theory, for the ordinal notation systems with
weaker signatures.

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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
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The categories of this paper are: q-bio
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: eess
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn