Interpreting true arithmetic in the theory of the r.e. truth table degrees (Q1902619)

Let \(\mathbf R_r\) and \(\mathbf D_r\) (\(\leq \text{\textbf{0'}}\)) be the partial orders of \(r\)-degrees of r.e. sets and \(r\)-degrees of sets \(r\)- reducible to \(\emptyset'\), respectively, where \(r \in \{T, wtt, tt, m\}\). It is well known that all the elementary theories of these structures are undecidable. The next more interesting question is, what is the complexity of these theories. \textit{L. Harrington} and \textit{R. A. Shore} [``Interpreting arithmetic in the theory of the r.e. Turing degrees'', unpublished]\ proved that the complexity of the theory of \(\mathbf R_T\) (in short \(\text{Th}(\mathbf R_T)\)) is just the same as that of true first-order arithmetic, i.e. they are \(m\)-equivalent. The same result for \(\mathbf D_T\) (\(\leq \text{\textbf{0'}}\)) is also true as shown by \textit{R. A. Shore} [J. Lond. Math. Soc. II. Ser. 24, 1-14 (1981; Zbl 0469.03027)]. Going through an interpretation of arithmetic in an auxiliary structure, the paper under review shows that true first-order arithmetic is also \(m\)-reducible to \(\text{Th}(\mathbf R_{tt})\), hence they are \(m\)- equivalent too. That is, the complexity of \(\text{Th}(\mathbf R_{tt})\) is just same as that of true first-order arithmetic.