Ask YC: What is your favourite paradox?
Mine has to be Olber's Paradox which asks 'Why is it dark at night?' Possible answers to this question are profound: the universe is of finite size or has a finite age.
This has always blown my mind. From Wikipedia: "The Banach–Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into several non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball."
Trippy.
Pretty much everything that involves abusing the Axiom of Choice ends up trippy and fun.
"The Axiom of Choice is obviously true, the Well-Ordering Principle is obviously false, and who can tell about Zorn's Lemma?"
Of course, AOC, WOP, and ZL are all logically equivalent.
[1] Possibly the nerdiest thought my brain has ever formed; this site makes me very happy.
Also, I have trouble with "incrementally" understanding this. I can understand that the limit of 1/n (n->INF) is 0, because even with large finite numbers we're getting close to 0, but no matter how many times I cut up an orange I detect no Banach-Tarski strangeness.
I think it's interesting to see the connection between the AC and the Law of Excluded Middle which also causes paradoxical issues from time to time.
http://en.wikipedia.org/wiki/Axiom_of_choice#Law_of_the_excl...
http://en.wikipedia.org/wiki/Zeno's_paradoxes
Those ancient Greeks knew their shit.
Sooo applicable to software development. I love it so much, I wrote a blog post about it.
Some mathematicians pull out calculus to "disprove" the paradox, but to me it disproves nothing. For example, you can show, mathematically, that sum( 1/(2^N) ) = 1 as N goes from 1 to infinity. The problem is that you have to go to infinity before it will sum to 1. If space is infinitely divisible and a particle has to traverse an infinite number of subspaces in order to move just a nanometer, I still don't see how it's possible that it could move at all.
love: it illustrates a fundamental limitation of democratic processes
hate: it's the main reason the IMO best known voting method (http://en.wikipedia.org/wiki/Condorcet_method) is difficult to explain/advocate for...
http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_...
For example, volume grows as the cube of linear dimension, but surface area only as the square. So as animals get bigger they have trouble radiating heat.
http://www.paulgraham.com/ycombinator.html
Perhaps large animals should make themselves into the shape of Gabriel's Horn.
Of course then they'll radiate too much heat.
Proof: Let's say we do not agree about anything. After thoroughly exploring our belief space and noticing this, we will have to agree that we do not agree on anything. So there you have it, a common belief.
BTW, I am looking for prior art on this one. Any philosophy students around? I think it is related to Russel's Paradox. Any help in elucidating this would be apreciated. Hey, we could co-author the publication :-P
"In fact, even if we allow an uncountable number of different colors for the hats, the axiom of choice provides a solution that guarantees that only finitely many prisoners must die."
This is by far the craziest result coming out of AOC.
"If this sentence is true, then Santa Claus exists."
After all, besides the importance to us, the earth is still a rather small dot in this universe and I really like to know what's happening out there. I have some hope that we can answer that paradox some day and maybe we will even get the first hints to that within my lifetime.
And perhaps wonder why we have "heating stations" all over the planet, when we really have a lack of energy.
The eery thing about the Fermi paradox is that it may spell doom for our own longevity.
The statement "My favorite paradox is the one I like least" has nothing paradoxical about it. It can easily be deduced to be false. Its just a contradiction, like saying "This sentence is true and false". Compare that to "This sentence is false" which is a paradox.