Let $\mathbb{C}_{exp}$ be the theory of the complex numbers in the language of exponential rings. I am interested in the Turing degree of $\mathbb{C}_{exp}$. As the natural numbers are definable in this setting, true arithmetic is coded in this theory so its Turing degree must at least be the $\omega$th iteration of the Turing jump - but is that sufficient?
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4$\begingroup$ It’s easy to see that the structure expanded with a predicate for $\mathbb R$ is equivalent to full second-order arithmetic. Unless I am mistaken, it is still open whether $\mathbb R$ is (parametrically) definable in $\mathbb C_{\exp}$, hence the answer most likely depends on properties of the theory that are beyond current knowledge. $\endgroup$– Emil JeřábekMar 31, 2015 at 14:09
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$\begingroup$ Good point. In particular then, the definability of $\mathbb{R}$ in $\mathbb{C}_{exp}$ would apparently imply that the recursive complex numbers (x+iy with x,y recursive reals) is not an elementary submodel of $\mathbb{C}$ (in the language of exponential rings). Is it known that it isn't? $\endgroup$– M CarlApr 1, 2015 at 12:41
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1$\begingroup$ It’s not known to me at any rate. It actually looks plausible to me that the recursive complex numbers are an elementary submodel, but I’m not really an expert on these matters. $\endgroup$– Emil JeřábekApr 1, 2015 at 13:32
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