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In Wikipedia the hyper-transcendental numbers are defined as complex numbers that are not the value, at an algebraic point, of a function which is the solution of an algebraic differential equation with integer coefficients and with algebraic initial conditions. But there are not given examples of such numbers, and it not specified if such numbers are computable or not.

I can suppose that such numbers can be the values of a hyper-transcendental function (as the gamma function) at some point, but how can we proof that such a number is not the value also of some elementary transcendental function?

Searching on the web I have not found much about such numbers. Are we sure that they exists? There is some known example?

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    $\begingroup$ But I'm asking about computable hyper-transcendental numbers, and computable real numbers are countable. $\endgroup$ – Emilio Novati Jan 10 '18 at 21:42
  • $\begingroup$ My first thought would be to look at Liouville numbers. They are computable and transcendental. I have no idea how to prove they are not solutions to differential equations of the sort you say, but it seems like they might not be. $\endgroup$ – Ross Millikan Jan 10 '18 at 21:50
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    $\begingroup$ If there is an algorithm to enumerate all non-hypertranscendental numbers in a computable way, you can compute a hypertranscendental number by diagonalization. $\endgroup$ – Taneli Huuskonen Jan 10 '18 at 22:14
  • $\begingroup$ @TaneliHuuskonen: but a diagonalization can prove the existence not the computability ? Or not? $\endgroup$ – Emilio Novati Jan 12 '18 at 8:23
  • $\begingroup$ The diagonalization method is constructive, so given a computable sequence of reals, you can compute a real not occurring in the sequence. $\endgroup$ – Taneli Huuskonen Jan 12 '18 at 9:31

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