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."
You can also deconstruct and rebuild that ball into another one of infinite size, I believe. I'm not sure if you can do infinite off the top of my head, but I'm certain of at least arbitrary.
Pretty much everything that involves abusing the Axiom of Choice ends up trippy and fun.
I'd really like someone to explain Banach-Tarski to me in some way that I can sort of understand. I understand it has to do with AOC, which seems totally reasonable, and then I'm lost.
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 honestly have a difficult time understanding the AC paradoxes, too. I recognize that it's totally necessary for many things (i.e. multidimensional integrals) and when it's consistent with a logical conclusion I have little trouble "seeing" it. Otherwise, though, not so much.
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.
Yes, the Condorcet vote is nice. But for practical purposes I've settled with approval voting as a favorite since it is easy to understand and can not be gamed.
Zeno is still my favorite. To summarize in modern terms: in order for a particle to move 1 inch, it must first move 1/2 inch. Before it can move 1/2 inch, it must move 1/4 inch. Before it can move 1/4 inch, it must move 1/8 inch. And so on. The paradox is, if a particle has to traverse an infinite number of infinitely small spaces before it can move even one inch, how is it possible that it can move at all?
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 Zeno's paradoxes. And apparently they aren't fully resolved today -- is the universe quantized? Is spacetime really continuous, or is that an unrealistic mathematical abstraction?
It's not really paradoxical [1], but I've always thought Gabriel's Horn was pretty cool. It's a (non-fractal) object with finite volume and infinite surface area.
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.
I read the wiki article and it turns out the horn uses precisely that dimensional difference in a very clever way. The radius varies as 1/x along the x-axis, so the volume grows as 1/x^2 while the surface area grows as 1/x. This means the growth rate of volume goes to zero much faster than that of the surface area. As a result the volume integral converges, while the surface area diverges.
Mine: People cannot completely disagree: or, we will always agree about at least one fact.
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.
Mine too! It's also the inspiration for my favorite math joke:
Consider the set of all sets that have never been considered...Wait, nevermind, they're gone!
This one is particularly cool, as on the face of it, the sentence just seems totally harmless. You think "well the sentence is not true and Santa Claus doesn't exist" and there's no hint of paradox.
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.
People often say the problem is the vast distances in space. I think they're wrong: a few thousand years is probably enough to colonize hundreds of worlds with generation ships. The main problem is the vast distance in time. An inhabitable planet may exist 20 light years from here, but odds are it is hundreds of millions of years behind or ahead of us in terms of evolution.
The eery thing about the Fermi paradox is that it may spell doom for our own longevity.
I think they are still listening to our internets and getting confused as to why we have a global network which seems to exist (at least 60% exist) so we can offer ourselves genital enlargements.
And perhaps wonder why we have "heating stations" all over the planet, when we really have a lack of energy.
That statement is false if you're using Merriam-Webster's definition of "lie" ("to make an untrue statement with intent to deceive"). I wrote that with intent to amuse and not with intent to deceive.
That's the definition of lie as a verb. I used lie as a noun. I hope you didn't think I was accusing you of lying. All I meant was that the statement was technically a lie, so the humor fell a little flat.
Yes, the interesting number paradox is particularly weak. But even it has an argument that seems puzzling initially, and has a much stronger version (called the Berry paradox, that has deep connections to Godel's incompleteness theorems).
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.
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.