| You give me a rational candidate for √2 and by the argument of the submitted article I “construct” another rational candidate with strictly smaller numerator and denominator. You give me a rational candidate p/q for log 2 and by the argument of the article I “construct” that p = q = 0. In both instances I create a contradiction thus showing that you could not have provided me such a rational to begin with. Now let me show you non-constructively that there are irrationals a and b with a^b rational. To prove this I consider c = √2^√2. I don’t know whether this is rational or irrational so I invoke the law of the excluded middle. Suppose c is rational. Then take a = b = √2 and we have a^b = c where a and b are irrational and c is rational. Otherwise suppose c is irrational. Then take a = c and b = √2 and we have a^b = (√2^√2)^√2 = √2^2 = 2 and again we have our result. What is the difference here? I have no idea whether c is rational or irrational. Either way I can make it work but I can’t tell you which possibility is the genuine one. You can actually make better choices of a and b and show this result constructively. This leads to a deeper question. Are there statements with no constructive proof and how do you prove that. This is beyond my expertise, but there are indeed high-powered logical tools that allow you prove such results for certain statements. The stuff that human beings have done by simply looking at the stars and dreaming really hard is pretty awe-inspiringly incredible. |