Timeline for Nontrivial theorems with trivial proofs
Current License: CC BY-SA 3.0
22 events
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| Dec 26, 2020 at 2:43 | comment | added | Ira Gessel | Here's another form of the pigeonhole principle: if $a_1+\cdots+a_n>b_1 + \cdots + b_n$ then $a_i>b_i$ for some $i$. The proof is that the contrapositive is trivial. (The "usual" form of the pigeonhole principle is the case in which $b_i=1$ for all $i$.) | |
| May 18, 2017 at 10:46 | comment | added | Joel David Hamkins | @Servaes I find it valuable and important to give names to one's fundamental mathematical principles and to understand when they are serving as axioms or whether they are reducible to still more fundamental principles. | |
| May 17, 2017 at 21:08 | comment | added | user75451 | @JoelDavidHamkins I recall first hearing of the pigeonhole principle when I was a graduate student. I found it absolutely ridiculous to give something so patently obvious a name, and found it even more absurd to write down a proof. I still feel this way. | |
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| May 8, 2014 at 21:54 | history | edited | François G. Dorais | CC BY-SA 3.0 |
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| May 8, 2014 at 21:54 | comment | added | François G. Dorais | (ACL, I've protected the question, which is a better solution than your edit, which I will now remove. I will simply lock the answer if the problem persists.) | |
| May 8, 2014 at 19:56 | history | edited | ACL | CC BY-SA 3.0 |
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| Jul 24, 2013 at 3:41 | comment | added | Joel David Hamkins | One usually proves the pigeonhole principle---in the form of the assertion that there is no injection from a natural number $n$ to a smaller natural number $k$---by induction on $n$. It is clearly true for $n=0$; if true at $n$, and we have an injection of $n+1$ to some smaller $k+1$, then by swapping two points we can produce an injection from $n$ to $k$. It is an interesting result that in very weak formal systems, one can separate the weak pigeonhole principal that there is no injecton from $2n$ to $n$ from the stronger assertion that there is no injection from $n+1$ to $n$. | |
| Jun 19, 2011 at 14:35 | comment | added | Quinn Culver | @gowers @Rod, I think it is a good example. Isn't its proof just contraposition? | |
| May 27, 2011 at 3:01 | comment | added | Selene Routley | Agreed. I don't even know a proof, or where to find one. | |
| May 25, 2011 at 18:09 | comment | added | gowers | Some might call that a trivial theorem with a non-trivial proof ... | |
| May 25, 2011 at 6:07 | history | answered | ACL | CC BY-SA 3.0 |