Could you tell if the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ is undecidable in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ ?
How can we prove the (un)decidability?
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1$\begingroup$ See the OP's related question at mathoverflow.net/q/224046/1946 $\endgroup$– Joel David HamkinsCommented Nov 26, 2015 at 2:55
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1$\begingroup$ The way you ask the question suggests that you somehow know already that it is undecidable (since you ask how we could prove this, rather than whether it is true). But is that right--are you claiming to know already that it is undecidable? If not, I'd suggest editing the question to use a different wording. $\endgroup$– Joel David HamkinsCommented Nov 26, 2015 at 3:20
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1$\begingroup$ Do you want the language to include an "evaluation" operation? $\endgroup$– Noah SchweberCommented Nov 26, 2015 at 6:34
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1$\begingroup$ He means: "is there an operation $ev$ which allows us to set the value of $x$" so that for example we can do $ev(e^x, 4)$ which then equals $e^4$? $\endgroup$– Andrej BauerCommented Nov 26, 2015 at 11:03
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3$\begingroup$ @MaryStar: I am not sure what you mean by "have to". What you have to do is state your problem(s) in precise terms. The point of my (and other people's) comments is that you get different problems by including (or not) some sort of evaluation function in your language. All these problems may be interesting. $\endgroup$– Laurent Moret-BaillyCommented Nov 29, 2015 at 9:07
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