Blue Solves Some Paradoxes

For thousands of years mathematicians and philosophers have spent countless hours attempting to understand any number of strange and fascinating problems.  Thanks to the limitless imagination of the human mind, we can create a certain kind of particularly difficult problem:  paradoxes that make sense only within the carefully defined laws of the puzzle itself.  Thousands upon thousands of hours have been spent solving these paradoxes, mostly because they're fun.  I don't think we'll ever break the United States' dependence on foreign oil by solving Curry's Paradox, will we?  But we can make an afternoon out of it.  This is just yet another extension of the human mind's beautiful insanity that I love so much.

As a note:  paradoxes are not unexpected outcomes - they are problems that seem to have no single outcome that is particularly more "logical" than the other.  Some people seem to think that the fact that most eco-friendly legislation results in companies actually increasing pollutants is a paradox.  They're wrong.  That's unexpected and unintuitive - ironic definitely - but not a paradox.  There's only one logical outcome.  Instead we're talking about logic games that have confounded mankind for years.  Problems that by design have no solution - or merely seem to.

However, since I'm just that brilliant, I'm going to go ahead right now to save you all a lot of trouble and solve some of our favorite paradoxes here on Planet Blue.  But I'm going to use a very special tool:  common sense.  Most paradoxes depend upon a very confusing and complicated form of rules, somehow always represented through a mathematical formula.  Well, I'm not going to even add 2 + 2 here.  Instead its all going to be through nothing but premium unleaded brilliance:


Barber Paradox

A town far far way has a very strange law:  every man must be clean-shaven.  But it goes further than that:  there is the sole barber in the town that is required by law to shave only the men who cannot shave themselves.  So then, can he shave himself?

The problem arises here because the barber can only shave men who cannot shave themselves.  He is capable of shaving himself, so he cannot shave himself.  But by law he must be clean-shaven, so he has to shave himself.  So what does this poor barber do?  Well, I would personally move because a town with such draconian facial hair laws cannot possibly be a pleasant place to live.  But the answer is actually incredibly simple:  he drives over to the next town and gets that town's barber to shave him.  If the town had a declared the barber to have a monopoly on shaving this solution wouldn't exist, but they forgot to.  So solved.

Unexpected Hanging Paradox

A prisoner sits on death row. The town sheriff comes in and tells the prisoner on Saturday that he will be hanged next week, and the prisoner will not know what day he will be hanged - it will be a surprise. The prisoner, desperate to guess when his time will come, rules out the next Saturday, because by that day every single other option will be excluded and so he will know for certain that he will be hanged that day and cannot possibly be surprised.  However, since Saturday is out, the prisoner then rules out Friday, since by that day every option will be excluded and so he could not possibly be surprised that day either.  Through this way he rules out every single day of the week and decides that he cannot possibly be hanged next week, much to his relief.  On Wednesday he is hanged, and he is very surprised.  How was he surprised?

This exact line of thinking occurred to me back in sixth grade when my teacher told the class that we will have a pop quiz next week and we won't know the day.  After about a week of pondering this issue, the quiz came on Thursday and I had forgotten to study.  Lucky it was an easy quiz and I passed.

Anyway, this paradox is really simple:  the prisoner is a fool.  There were only two things that could have happened on every day of the week, so he should have been prepared for either one:  living or dying.  Only having two possible outcomes on each day isn't really much of a surprise.  The entire line of thought was just wishful thinking, and he should have spent his last few weeks getting his affairs in order and picking out how he wanted his last steak to be cooked.

The Ball and Vase Paradox

You have a vase and lots of balls.  At one minute to noon, you put in ten balls and remove one.  At thirty seconds to noon, you put in ten more balls and remove another.  At fifteen seconds you repeat.  At 7.5 seconds you repeat again.  At 3.75 seconds you repeat again.  You continue the steps infinitely, each time halving the time between steps.  When the clock strikes twelve, how many balls are in the vase?

This one is an admittedly complex math problem.  If you were to perform the arithmetic endlessly you'd find that there are infinite balls in the vase.  If you were to use calculus you'd find there are no balls in the vase:  the number of balls you add is infinite and the number of balls you remove is infinite.  (∞ - ∞ = 0)  So which is it?

Actually simulating this paradox in real life is a little more complex than the last mostly because it requires a lot of dexterity on your part since you need to perform an infinite number of actions in an infinitely small amount time.  I suggest you sleep well the night before.  Anyway, the answer is pretty obvious:  the vase breaks.  Just how many balls can you fit inside a vase, even a big vase?  You're effectively putting in an infinite number of objects in a limited space.  That poor vase is not going to last long.  So by noon the vase will be nothing but shattered porcelain on the ground, and asking how many balls it holds is a meaningless question, since technically there will no longer be a vase.

Achilles and the Tortoise Paradox

The great hero Achilles begins a race with a slow little tortoise.  Achilles can cover ten feet in the time that it takes the tortoise to cover one.  At the start of the race, to make things more sporting, Achilles gives the tortoise a ten foot head-start.  When Achilles covers ten feet, the tortoise cover one.  When Achilles covers one foot, the tortoise covers 1/10th of a foot.  When Achilles covers 1/10th of a foot, the tortoise covers 1/100th, and so on.  How can Achilles possibly pass the tortoise?

This paradox is a little difficult to replicate because it would be rather difficult to find Achilles, let alone try to get him to run a race since he's a little bit... dead.  Scholars have spent thousands of years trying to deduce the reason why this paradox is false (since it clearly is or else you'd never be able to pass any moving object ever).  Well here's the major problem here:  neither Achilles or the tortoise can move into spaces that are infinitely small.  It would be a little difficult to cover just 1/1000th of a foot, let alone 1/1000000000th.  Unfortunately for this paradox, an animal foot is a little too big to cover such tiny spaces, even if you convinced both Achilles and his amphibian friend to purposefully try to cover as small of ranges as possible while they're supposed to be racing.  There's a lower limit here, so it can never get infinitely small, so passing the tortoise is as simple as well... passing a tortoise.  Eventually Achilles would simply step right over the tortoise and go on to win the race.

But as seen with the tortoise's many races with the hare, the tortoise is a cheater, so Achilles may have problems beyond even what Zeno imagined here.

Interesting Number Paradox

A mathematician, spending quite a lot of time on Wikipedia, finds that just about every real number is the first member of a set of numbers with special properties: first whole number, first even number, first prime number, first cardinal number, etc. So he then decides to make a new set: uninteresting numbers, numbers which belong to none of these sets. However by assigning a certain number with the title of "first uninteresting number" it suddenly becomes interesting. The second uninteresting number is now the first uninteresting number, and now it is interesting. The mathematician is now stuck with a problem: can there ever be an uninteresting number?

The answer is "yes".  The number 1142 is extremely boring and no category of "uninteresting number" can ever make it interesting under any circumstances.  1142 is like a plain American cheese sandwich on whole wheat bread:  you eat it, you digest it, but you never care.  Its the most boring thing in the universe.  By extension, five is the most interesting number in the world, and even if it were in the category of "uninteresting numbers" it still would be massively interesting.

Call it a cop-out, but remember this is a paradox of "interestingness" - a highly subjective quality.  I find this article I'm writing here to be fascinating, you might have given up reading in the first paragraph.  Its all about opinion, and I say 1142 is uninteresting no matter what number set it may be a part of.  Saying a number is "the first uninteresting number" doesn't make it interesting, nor does calling it the 1142nd uninteresting number.

Also some people find all numbers to be uninteresting already.

Omnipotence Paradox

God is all-powerful being and can do anything. So She* creates a rock so heavy that even She cannot lift it. Is this possible?

This is an interesting problem, that seemingly is a tight bind for poor God.  Seemingly Her omnipotence is impossible here:  She can't make the rock or she can't lift it, either way omnipotence is lost since this means there is an action God cannot perform.  C.S. Lewis called this entire paradox "nonsense" but rejecting the very nature of the question is against the rules.  You have to accept the paradox's rules in order to solve it.

Surprisingly, C.S. Lewis of all people just wasn't creative enough to solve this problem.  (I'm as shocked as you are.)  Its really simple really.  All God needs to do is make a rock of any heaviness, and then divide Herself in two, a lesser and a greater form.  The lesser cannot lift the rock, thus meaning that God has made a rock she cannot lift.  The greater can lift the rock, thus meaning that God can lift even a rock She cannot lift.  In this way God has performed both actions at once:  simultaneously lifting and not lifting the rock.  She is omnipotent after all so this isn't all that silly.  Solved.

Duh.

ツ

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* Call me a commie liberal piece of shit, but I reject your masculine pronouns for your deity and insert my own. If you've ever wondered just what I think God's form is since I so often have problems with how other people perceive him/her/it, just imagine Rosalina from "Super Mario Galaxy" and add wings.