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A note on definability in fragments of arithmetic with free unary predicates
We carry out a study of definability issues in the standard models of Presburger and Skolem arithmetics (henceforth referred to simply as Presburger and Skolem arithmetics, for short, because we only deal with these models, not the theories, thus there is no risk of confusion) supplied with free unary predicates — which are strongly related to definability in the monadic SOA (second-order arithmetic) without * or +, respectively. As a consequence, we obtain a very direct proof for $\Pi^1_1$-completeness of Presburger, and also Skolem, arithmetic with a free unary predicate, generalize it to all $\Pi^1_n$-levels, and give an alternative description of the analytical hierarchy without * or + . Here ‘direct’ means that one explicitly m-reduces the truth of $\Pi^1_1$-formulae in SOA to the truth in the extended structures. Notice that for the case of Presburger arithmetic, the $\Pi^1_1$-completeness was already known, but the proof was indirect and exploited some special $\Pi^1_1$-completeness results on so-called ‘recurrent’ nondeterministic Turing machines — for these reasons, it was hardly able to shed any light on definability issues or possible generalizations.