We consider the constructive ordinal notation system for the ordinal
Ο΅0β that were introduced by L.D. Beklemishev. There are fragments of
this system that are ordinal notation systems for the smaller ordinals
Οnβ (towers of Ο-exponentiations of the height n). This
systems are based on Japaridze's provability logic GLP. They are
closely related with the technique of ordinal analysis of PA and
fragments of 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 β₯Ο4β, have undecidable elementary theories.
We also prove that the fragments of the full system, for ordinals
β€Ο3β, have decidable elementary theories. We obtain some results
about decidability of elementary theory, for the ordinal notation systems with
weaker signatures.Comment: 23 page