Comments on: Constructive gem: irrational to the power of irrational that is rational

By: Andrej Bauer

Andrej Bauer — Sat, 10 Dec 2011 10:08:20 +0000

@Terry: Hmm, actually I do not know what you mean if you have in mind the usual non-constructive proof involving $\sqrt{2}^\sqrt{2}$, because that number is transcedental, as it is the square root of the Gelfand-Schneider Constant.

By: Andrej Bauer

Andrej Bauer — Sat, 10 Dec 2011 08:34:08 +0000

@Terry: That’s a good point. Can we have a constructive proof of two irrational algebraic numbers whose power is rational?

By: Terry

Terry — Sat, 10 Dec 2011 01:40:56 +0000

Note though that the non-constructive argument proves something stronger than the constructive one, namely that there exist two irrational algebraic numbers $a$, $b$ such that $a^b$ is rational.

By: Cody Roux

Cody Roux — Mon, 05 Jul 2010 11:56:01 +0000

I see the proof of the above theorem in a somewhat different way than “a useless example of non-constructive proof”. Let me make my point using a Car Analogy. You go to a car dealer, and he supplies a car to you, along with a guarantee: He must supply a -working- car to you, if needed. If you never use the car, the contract is implicitly satisfied, otherwise, if you do try to start the car, then two cases may occur: The car starts, and everything is fine, or the car does not start. In this case, you make use of your guarantee, and the dealer supplies a new, working car (or some other car with another guarantee). You do the same for your classical proof of existence of 2 irrational numbers $a$ and $b$ such that $a^b$ is rational: act as if $a=\sqrt{2}$ and $b=\sqrt{2}$ satisfy your theorem. If anyone calls you out on it, by supplying a proof that $\sqrt{2}^\sqrt{2}$ is irrational, you can “backtrack” by supplying new witnesses to your theorem: $a=\sqrt{2}^\sqrt{2}$ and $b=\sqrt{2}$. In a sense, the non-constructive proof does contain “less information” than the constructive proof. However it does supply a “guarantee” as in the above sense, that is you may use your witnesses with the caveat of perhaps needing to substitute them for “correct” witnesses at a further point.

By: Ananymous

Ananymous — Thu, 11 Mar 2010 19:47:27 +0000

An intuitionist logic professor told me that once he was trying to solve a problem from his own book preparing for the next day’s undergrad logic lecture. After hours of unsuccessful thinking he noticed that he was not able to solve it because he was not using PEM. :)