Timeline for About the proof of the proposition "there exists irrational numbers a, b such that a^b is rational"
Current License: CC BY-SA 2.5
9 events
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| Mar 24, 2015 at 13:15 | comment | added | Andrej Bauer | I just feel like saying $(\sqrt{2})^{\log_2 9}$. But more seriously, it is unpedagocial to speak about what we know and do not know when explaining the form of a given proof. The proof in question is not constructive because it employs the inference rule of excluded middle -- and the proof remains non-constructive no matter what else you prove or know. However, the proof can be transformed to a different proof which is constructive (assuming there is a constructive proof of irrationality of $\sqrt{2}^{\sqrt{2}}$). | |
| May 22, 2011 at 8:12 | vote | accept | day | ||
| Apr 25, 2011 at 1:56 | comment | added | Pete L. Clark | @Maxime: in fact, I have assigned this problem (or rather a slight generalization -- Schuh's Divisor Game) as homework, but in a higher level course. | |
| Apr 25, 2011 at 1:20 | comment | added | Maxime Bourrigan | @Carl, Pete: I think that the strategy-stealing proof that the first player has a winning strategy at Chomp (en.wikipedia.org/wiki/Chomp) is also a good example to convince students of this fact. | |
| Apr 24, 2011 at 22:10 | comment | added | Pete L. Clark | Carl's remark is right on the money. I have used exactly this example in a sophomore-level introduction to proofs course for exactly this purpose. (Unfortunately the most common reaction was utter confusion...) | |
| Mar 2, 2011 at 15:23 | comment | added | Carl Mummert | I think the reason the $\sqrt{2}$ case is interesting is that the proof that $\sqrt{2}$ is irrational is very easy, and taught to young undergraduates. But the Gelfond–Schneider theorem is much more advanced. So for a student who is not very advanced, this gives a nice example of an existence proof where the student isn't in possession of a proof that any particular number is a witness. That's a nice pedagogical tool to develop students' intuition that the proof of an existence statement does not have to construct the thing that is shown to exist. Examples like $e^{\ln(2)} = 2$ don't do that. | |
| Feb 28, 2011 at 23:01 | comment | added | Andy Putman | @David : The second one is the rational one. en.wikipedia.org/wiki/Gelfond–Schneider_theorem | |
| Feb 28, 2011 at 22:43 | comment | added | David Roberts♦ | But we do know which of the two cases is irrational - I suppose the question is how constructive is the proof of that fact? | |
| Feb 28, 2011 at 22:37 | history | answered | Andreas Blass | CC BY-SA 2.5 |