

Ask YC: What is your favourite paradox? - moog

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.
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manny
Banach-Tarski paradox: <http://en.wikipedia.org/wiki/Banach–Tarski_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.

~~~
tel
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.

~~~
mnemonicsloth
My favorite quote about the Axiom of Choice[1]:

"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.

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ghiotion
Zeno's paradox, which proves that motion is an illusion:

<http://en.wikipedia.org/wiki/Zeno's_paradoxes>

Those ancient Greeks knew their shit.

~~~
mattmaroon
In that same article you linked it explains why those paradoxes don't prove
shit.

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ghiotion
True. Perhaps I should have put quotes around proves. Still, I remember what a
big impact Zeno had on me as a young philosophy student.

~~~
eru
You should have studied math. ;)

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TheWama
I have a love/hate relationship with Condorcet's paradox
(<http://en.wikipedia.org/wiki/Condorcet's_paradox>):

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...

~~~
eru
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.

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paulgb
I find Hilbert's Paradox is handy when thinking about infinity.

[http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_...](http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel)

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Zeromus
I didn't have one, but thanks to this article I'll be wasting the next few
days reading this: <http://en.wikipedia.org/wiki/List_of_paradoxes>

~~~
corroded
you didn't have to drag us in with you T_T thanks to you i'll be also reading
that list along with all the ones that everyone posted here XD

~~~
eru
Just to make you procrastinate even more:

<http://en.wikipedia.org/wiki/List_of_cognitive_biases>

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mnemonicsloth
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.

<http://en.wikipedia.org/wiki/Gabriel%27s_Horn>

~~~
tlrobinson
Finite volume and infinite surface area, eh?

 _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.

~~~
mercurio
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.

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ovi256
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

~~~
eru
I can always refuse to argue with you and punch you instead.

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yelsgib
[http://en.wikipedia.org/wiki/Prisoners_and_hats_puzzle#Count...](http://en.wikipedia.org/wiki/Prisoners_and_hats_puzzle#Countably_Infinite-
Hat_Variant)

"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.

~~~
eru
The Axiom of Choice saves lives!

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mercurio
Russell's paradox.

<http://en.wikipedia.org/wiki/Russell's_paradox>

~~~
imgabe
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!

~~~
jsmcgd
Mine too too. It makes logic not look so logical anymore.

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dfranke
The hangman paradox: <http://en.wikipedia.org/wiki/Unexpected_hanging_paradox>

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yters
Curry's paradox is cool:

"If this sentence is true, then Santa Claus exists."

<http://plato.stanford.edu/entries/curry-paradox/>

~~~
mercurio
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.

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coffeeaddicted
The Fermi paradox - "where are they?"
<http://en.wikipedia.org/wiki/Fermi_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.

~~~
wlievens
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.

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simanyay
Grandfather paradox: <http://en.wikipedia.org/wiki/Grandfather_paradox>

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__
The one I like least.

~~~
mercurio
Not to rain on your parade, but your statement is a lie and not a paradox.

~~~
__
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.

~~~
mercurio
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.

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tlrobinson
Schrödinger's cat

~~~
bayareaguy
It's not a paradox by itself, but the Heisenburg uncertainty principle leads
to a lot of unintuitive things too.

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aswanson
Existence.

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doubleplus
The 20 Eye: <http://www.zappos.com/images/725/7257495/5089-291353-d.jpg>

~~~
eru
?

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mixmax
girls

~~~
kirubakaran
If they are a paradox to you, my hero who scored with a porn star, I don't
know what to say :-)

~~~
mrtron
If that is your hero definition, what is your superhero definition?

~~~
kirubakaran
mixmax wearing underwear over his pants, duh

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anewaccountname
Actually, Olber's Paradox can also be answered by assuming the Universe is
fractally distributed.

------
bootload
python ~ <http://www.paulgraham.com/pypar.html>

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joseakle
Is "Don't listen to me." a paradox?

~~~
eru
No. I read it and followed your instructions. It's not even a contradiction
when written down.

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wallflower
"Do as I say and not as I do."

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RNlR
Democracy or mob-intelligence

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dkokelley
This is a false sentence.

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michaelneale
'fer me ?

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martianpenguin
<http://en.wikipedia.org/wiki/Paradoxical_combinator>

