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Does there exist any rubric where provably transcendental real numbers emerge, in a meaningful way, as rare among all the transcendental numbers?

Here are some of the things I'm worried about:

1) To talk about provably transcendental numbers, it seems only fair to consider them as a subset of some sort of set of definable real numbers (relative to some appropriate language). If the language is countable, that means comparing two countable sets, so measure-theoretic language doesn't seem to help.

2) Some transcendentality proofs naturally apply to all the numbers in a definable uncountable set (which of course contains many undefinable numbers). Small variations of Liouville's famous original construction yield uncountable sets of transcendentals. So a countable language that can only encode a countable number of proofs can still establish the transcendentality of more than countably many numbers.

Perhaps something like this: relative to a fixed language one can define a complexity for definable transcendental reals by the length of their shortest defining formula. Among those of a given complexity, some fraction admit transcendentality proofs. Perhaps this fraction must go to 0 with the complexity for any reasonable theory? (This seems to me a meaningful question despite that attendant undecibilities concerning whether a formula defines a numbers, whether two formulas define the same number, etc.)

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  • $\begingroup$ I do see that this question mathoverflow.net/questions/26402/… has a kinship with mine, but the questions don't appear equivalent. $\endgroup$
    – David Feldman
    Commented Mar 26, 2011 at 7:46
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    $\begingroup$ I am not sure if such a theory can be fruitful... even Thue-Siegel-Roth, one of the most powerful results in Diophantine approximation, is a very weak test in transcendence testing and is also non-effective. Baker's methods give better transcendence tests but are still very weak. Given these obstructions, I am not sure what other ideas can be made available to rigorously separate the 'testable' from the non-testable, since we expect the set of testable numbers to be much larger than what is known to be testable today. $\endgroup$
    – Stanley Yao Xiao
    Commented Mar 27, 2011 at 0:12
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    $\begingroup$ Since student days I have the following dream: In the realm of decimal expansions the expansions of rational numbers are distinguished by being periodic, and in the realm of continued fractions the expansions of rationals are finite and the expansions of quadratic irrationals are periodic. Someone should come up with an encoding of the reals that imparts algebraic numbers of arbitrary degree some distinguished feature. All "expansions" not having this feature would then automatically encode a transcendental number. $\endgroup$
    – Christian Blatter
    Commented Mar 27, 2011 at 9:02

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I am not sure this is what your question aims at, but: $e^{1/n}$ (see Wikipedia) for natural n> 0, are transcendental and have very simple continued fraction expansion. If memory serves, Hurwitz has theorems generalizing this phenomenon (and perhaps Baker, much later). Sorry to be so shaky on the details here... The characterization, if memory serves, is again in terms of continued fractions where the pattern of the coefficients follows some arithmetic series. In that sense, it is a countable set of provably transcendental real numbers, that is nonetheless very rare among all transcendental numbers.

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