We prove that the injectively omega-tree-automatic ordinals are the ordinals smaller than $omega^{omega^omega}$. Then we show that the injectively $omega^n$-automatic ordinals, where $n>0$ is an integer, are the ordinals smaller than $omega^{omega^n}
$. This strengthens a recent result of Schlicht and Stephan who considered in [Schlicht-Stephan11] the subclasses of finite word $omega^n$-automatic ordinals. As a by-product we obtain that the hierarchy of injectively $omega^n$-automatic structures, n>0, which was considered in [Finkel-Todorcevic12], is strict.
We consider $omega^n$-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length $omega^n$ for some integer $ngeq 1$. We show that all these structures are $omega$-tree-aut
omatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for $omega^2$-automatic (resp. $omega^n$-automatic for $n>2$) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for $omega^n$-automatic boolean algebras, $n > 1$, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a $Sigma_2^1$-set nor a $Pi_2^1$-set. We obtain that there exist infinitely many $omega^n$-automatic, hence also $omega$-tree-automatic, atomless boolean algebras $B_n$, $ngeq 1$, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [FT10].
An $omega$-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for $omega$-tree-automatic structures. We prove first that th
e isomorphism relation for $omega$-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for $omega$-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is neither a $Sigma_2^1$-set nor a $Pi_2^1$-set.
We study the problem of the existence of unconditional basic sequences in Banach spaces of high density. We show, in particular, the relative consistency with GCH of the statement that every Banach space of density $aleph_omega$ contains an unconditional basic sequence.
