Talk:Gelfond–Schneider constant

What's that nonconstructive proof thing really trying to say?

e^{ln 2} after all is an irrational number to an irrational power. Phr 06:05, 18 February 2006 (UTC)

The square root mentioned in the article is irrational, but raised to the power √2 you get 2, which is rational. This shows that √2^√2 is transcendental but seems to me like a constructive proof that an irrational to the power of an irrational can be rational, so I don't know what the nonconstructive proof referred to. Chenxlee (talk) 22:17, 6 February 2008 (UTC)

The Pythagoreans knew the square root of two was irrational. Suppose in 1901 you (or Hilbert) did not know whether ${\sqrt {2}}^{\sqrt {2}}$ was rational or not: if it was rational then $\left({\sqrt {2}}\right)^{\left({\sqrt {2}}\right)}$ would have been an example of an irrational to the power of an irrational being rational; if it was irrational then $\left({\sqrt {2}}^{\sqrt {2}}\right)^{\left({\sqrt {2}}\right)}$ was an example of an irrational to the power of an irrational being rational. That was non-constructive as it depended on the status of a number without finding out that status. --Rumping (talk) 16:22, 9 June 2008 (UTC)

I still do not get it. How any of these shows that the number is transcendental? And what is so special about "an irrational to the power of an irrational being rational", which we know is not a contradiction on its own.81.6.34.246 (talk) 20:32, 12 November 2018 (UTC)