The fact that there exists irrational number $a,b$ such that $a^b$ is rational is proved by the law of excluded middle, but I read somewhere that irrationality of $\sqrt{2}^{\sqrt{2}}$ is proved constructively. Do you know the proof?

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    $\begingroup$ See this. $\endgroup$ – David Mitra Jul 18 '13 at 15:32
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    $\begingroup$ @DavidMitra: many thanks, I was not aware of that theorem. $\endgroup$ – Ron Gordon Jul 18 '13 at 15:43
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    $\begingroup$ One can find this result, and its proof, in Ivan Niven's book Irrational numbers. $\endgroup$ – David Mitra Jul 18 '13 at 15:46
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    $\begingroup$ Interesting question! Just about all proofs of transcendence I'm aware of essentially assume the target is algebraic and conclude that there is an integer strictly between $0$ and $1$. Not sure whether anything like this can be turned into a constructively (or intuitionistically) valid argument. I am also not aware of any proof of irrationality of $\sqrt2^{\sqrt2}$ other than the transcendence proof of Gelfond-Schneider. (Of course, I'm probably just overlooking something.) $\endgroup$ – Andrés E. Caicedo Jul 18 '13 at 15:48
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    $\begingroup$ @Andres: isn't the definition of "transcendental" just "not algebraic"? Then a constructive proof that a number is transcendental is precisely a proof of a contradiction from the assumption that number is algebraic. I am not very familiar with the actual proof so I can't say much about the details, but see cs.nyu.edu/pipermail/fom/2005-March/008846.html $\endgroup$ – Carl Mummert Jul 19 '13 at 11:44

Since this is a well-established result, this is a community wiki post.

Relevant question: Deciding whether $2^{\sqrt2}$ is irrational/transcendental

Kuzmin proved the following claim in 1930:

Theorem: If $\alpha\neq 0,1$ is algebraic, $\beta$ is positive and rational, not a perfect square, then $\alpha^{\sqrt{\beta}}$ is transcendental.

Unfortunately the paper is in Russian and I failed to find an English translation. A corollary of this is that $2^{\sqrt{2}}$ is transcendental, and so is its square root $\sqrt{2}^{\sqrt{2}}$.

The outlines of both Gelfond and Kuzmin's constructive proof can be found here.

As David Mitra pointed out the comments, Niven's book had a section dedicated to this. I love Niven's book so much. The technique is similar to the adapted proof I posted here, proof by contradiction.

Rough idea about the construction: First assuming $\alpha^{\sqrt{\beta}}$ is algebraic. Then using sufficient large degree Lagrange interpolation polynomial to approximate $e^{(\ln \alpha)x}$ at points $\{a+ b\sqrt{2}\}$ for $a,b\in \mathbb{Z}$. Let the number of points go to infinity the error will go to zero, this shows a transcendental function $\alpha^x$ can be interpolate using countably many algebraic points. Contradiction.

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    $\begingroup$ Not a constructive argument (in the sense of mathematical logic), though. $\endgroup$ – Andrés E. Caicedo Jul 19 '13 at 5:00
  • $\begingroup$ @AndresCaicedo Thanks for the heads up, I didn't realize there could be a constructive proof in the sense of mathematical logic. I appreciate if you write one, I want to learn it as well. :) $\endgroup$ – Shuhao Cao Jul 19 '13 at 5:07
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    $\begingroup$ I don't know if there is one. It seems a difficult question. $\endgroup$ – Andrés E. Caicedo Jul 19 '13 at 5:11
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    $\begingroup$ To all the voters out there: the OP asks whether the irrationality of ${\sqrt 2} ^{\sqrt 2}$ can be proved CONSTRUCTIVELY. I fail to see how this upvoted answer addresses the question. $\endgroup$ – Alex M. Jun 5 '17 at 20:50
  • $\begingroup$ @AlexM. This question shows up as the answerable one in a "duplicate of..." chain for a number of questions that were not asking for this requirement. Some of them weren't even asking about $\sqrt{2}^\sqrt{2}$. $\endgroup$ – GPhys Jun 19 '17 at 6:35