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What's your favorite Paradox?

Discussion in 'BlackHat Lounge' started by BlogPro, Aug 1, 2016.

  1. BlogPro

    BlogPro Jr. VIP Jr. VIP

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    Paradoxes have always fascinated me.

    Paradoxes are essentially self-contradicting statements or a bunch of statements that cannot ever come to a logical conclusion. History is riddled with paradoxes and many of them have stumped even the most profound of logicians.

    Paradoxes are built around the flaw of simpler login.It is a situation where a group of "reasonable" assumptions along with a group of "reasonable" rules of deduction lead to a contradiction of one of the assumptions. Generally this means that either one of the assumptions or rules of deduction wasn't so reasonable after all.

    Certain times, they can be proved mathematically - however logically they make no sense.

    Here are a few of my favorite ones -

    Please note, these are either written from memory or copied from various sources and have been posted as is, so as to maintain the original integrity and the cumulative meaning of the sentences. A few of them may also contain commentary from me that wasn't really a part of the original paradox, simply to make it simpler for people to grasp. Please excuse typos and brevity.

    The Achilles and The Tortoise Paradox

    The Paradox of Achilles and the Tortoise is one of a number of theoretical discussions of movement put forward by the Greek philosopher Zeno of Elea in the 5th century BC. It begins with the great hero Achilles challenging a tortoise to a footrace. To keep things fair, he agrees to give the tortoise a head start of, say, 500m.

    When the race begins, Achilles unsurprisingly starts running at a speed much faster than the tortoise, so that by the time he has reached the 500m mark, the tortoise has only walked 50m further than him. But by the time Achilles has reached the 550m mark, the tortoise has walked another 5m. And by the time he has reached the 555m mark, the tortoise has walked another 0.5m, then 0.25m, then 0.125m, and so on. This process continues again and again over an infinite series of smaller and smaller distances, with the tortoise always moving forwards while Achilles always plays catch up.

    The Paradox of the court, also known as the counter dilemma of Euthalos.

    Many years ago, a Law teacher came across a student who was willing to learn but was unable to pay the fees. The student struck a deal saying, "I will pay your fee the day I win my first case in the court".

    The teacher agreed and proceeded with the law course. When the course was finished and the teacher started pestering the student to pay up the fee, the student reminded him of the deal and pushed days.

    Fed up with this, the teacher decided to sue the student in the court of law and both of them decided to argue for themselves.

    The teacher put forward his argument saying:
    "If I win this case, as per the court of law, the student has to pay me as the case is about his non-payment of dues.
    And if I lose the case, the student will still pay me because he would have won his first case...
    So either way, I will get the money".

    Equally brilliant, the student argued back saying:
    "If I win the case, as per the court of law, I don't have to pay anything to the teacher as the case is about my non-payment of dues.
    And if I lose the case, I don't have to pay him because I haven't won my first case yet....
    So either way, I am not going to pay the teacher anything".

    This is one of the greatest paradoxes ever recorded.

    Who is right and who is the winner?

    This is part of ancient Greek history. The lawyer teacher was Protagoras (c.485-415 BCE) and the student was Euthalos.

    The Barber Paradox

    Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves.

    Under this scenario, we can ask the following question: Does the barber shave himself?

    Asking this, however, we discover that the situation presented is in fact impossible:
    - If the barber does not shave himself, he must abide by the rule and shave himself.
    - If he does shave himself, according to the rule he will not shave himself

    The Omnipotence Paradox

    So if God is omnipotent, he can create anything. The paradox lies in the absolute resolve of his power.

    Can he create an object that cannot be moved? If he's omnipotent, he can.

    Now, can HE move that object? Again, if he's omnipotent, he can.

    If he creates the object, but cannot move it. Is he still omnipotent? If he creates the object, but is able to move it? Again, is he still omnipotent?

    In simpler words, what would happen if an unstoppable force hit an immovable object?

    The Ship of Theseus Paradox

    Suppose there is a wooden Ship out on the Sea that does several small trips in a day while returning to the port at night. It is comprised of 150 wooden parts. Every night, the crew removes one old part, and puts them in a ship building yard, while replacing it with an exact same new part.

    This goes on for 150 days. After 150 days, there are two identical ships. One in the sea, another in the dockyard.

    Which is the original Ship of Theseus?

    The Time Travel Paradoxes

    The concept of time travel has fascinated people around the world and has given birth to several different paradoxes. Here are a couple of them.

    The Family Tree Paradox

    A baby girl is mysteriously dropped off at an orphanage in Cleveland in 1945. "Jane" grows up lonely and dejected, not knowing who her parents are, until one day in 1963 she is strangely attracted to a drifter. She falls in love with him. But just when things are finally looking up for Jane, a series of disasters strike.

    First, she becomes pregnant by the drifter, who then disappears.

    Second, during the complicated delivery, doctors find that Jane has both sets of sex organs, and to save her life, they are forced to surgically convert "her" to a "him."

    Finally, a mysterious stranger kidnaps her baby from the delivery room.

    Reeling from these disasters, rejected by society, scorned by fate, "he" becomes a drunkard and drifter.

    Not only has Jane lost her parents and her lover, but he has lost his only child as well.

    Years later, in 1970, he stumbles into a lonely bar, called Pop's Place, and spills out his pathetic story to an elderly bartender. The sympathetic bartender offers the drifter the chance to avenge the stranger who left her pregnant and abandoned, on the condition that he join the "time travelers corps." Both of them enter a time machine, and the bartender drops off the drifter in 1963. The drifter is strangely attracted to a young orphan woman, who subsequently becomes pregnant.

    The bartender then goes forward 9 months, kidnaps the baby girl from the hospital, and drops off the baby in an orphanage back in 1945. Then the bartender drops off the thoroughly confused drifter in 1968, to enlist in the time travelers corps.

    The drifter eventually gets his life together, becomes a respected and elderly member of the time travelers corps, and then disguises himself as a bartender and has his most difficult mission: a date with destiny, meeting a certain drifter at Pop's Place in 1970.

    The question is: Who is Jane's mother, father, grandfather, grandmother, son, daughter, granddaughter, and grandson?

    The girl, the drifter, and the bartender, of course, are all the same person. These paradoxes can made your head spin, especially if you try to untangle Jane's twisted parentage. If we draw Jane's family tree, we find that all the branches are curled inward back on themselves, as in a circle.

    We come to the astonishing conclusion that she is her own mother and father! She is an entire family tree unto herself.

    The Hamlet Paradox

    The Bootstrap Paradox is a paradox of time travel that questions how something that is taken from the future and placed in the past could ever come into being in the first place. It’s a common trope used by science fiction writers and has inspired plotlines in everything fromDoctor Who to the Bill and Ted movies, but one of the most memorable and straightforward examples—by Professor David Toomey of the University of Massachusetts and used in his book The New Time Travellers—involves an author and his manuscript.

    Imagine that a time traveller buys a copy of Hamlet from a bookstore, travels back in time to Elizabethan London, and hands the book to Shakespeare, who then copies it out and claims it as his own work. Over the centuries that follow, Hamlet is reprinted and reproduced countless times until finally a copy of it ends up back in the same original bookstore, where the time traveller finds it, buys it, and takes it back to Shakespeare.

    Who, then, wrote Hamlet?

    --

    I'll add more if this thread gains some attention. Hope you guys enjoy reading about them.
     
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  2. Sherbert Hoover

    Sherbert Hoover Jr. Executive VIP Jr. VIP

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  3. Conor

    Conor Elite Member

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    This one doesn't make sense. If Achilles can get to 500m by the time the tortoise reaches 550m, he'll easily overtake the tortoise eventually and win the race.

    Source: 5 years of owning a drivers license.
     
  4. BlogPro

    BlogPro Jr. VIP Jr. VIP

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    That is a beauty of the paradox. Logically it never makes sense, but it can be proved mathematically. It also relies on the fact that our mind cannot fathom the concept of infinity.

    The concept lies in the fact that when Achilles goes from 0 to 500, the tortoise will have moved a bit too. So when Achilles covers half that distance, the tortoise will seemingly move a bit ahead. Everytime Achilles covers half the distance between him and the tortoise, the tortoise moves a bit away, so and so forth unto infinity.


    One of my favorites too!
     
  5. blogzandstuff

    blogzandstuff Elite Member

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    i like this, is it the same broom?
     
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  6. Conor

    Conor Elite Member

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    It's intriguing but I'm still failing to see the paradox.

    If we assume Achilles and the tortoise maintain a constant speed throughout, then Achilles will overtake the tortoise eventually.

    Achilles: 1km/hour
    Tortoise: 100 meters/hour

    Tortoise gets to 500m in 5 hours
    Achilles gets there in 30 minutes
    Total time for Achilles to reach 500m after waiting for the tortoise: 5 hours, 30 minutes

    In that time, the tortoise would have moved 50 meters ahead. So, in 5 hours, 30 minutes, the tortoise would be at 550m, while Achilles would be at 500m. Fine.

    Now, lets say they run to the 600m mark. Achilles would get there in 6 minutes. The tortoise would take 30, because he's already at 550m.

    Tortoise time to 600m: 6 Hours
    Achilles time to 600m: 5 Hours, 36 Minutes
     
  7. davids355

    davids355 Jr. VIP Jr. VIP

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  8. BlogPro

    BlogPro Jr. VIP Jr. VIP

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    What you're doing is applying base logic here. Of course Achilles would win by a landslide. But then the points raised in the paradox are true as well. Achilles would have to take an infinite number of small steps to cover half the distance, while the tortoise keeps gaining.

    Perhaps this explanation should help you http://platonicrealms.com/encyclopedia/zenos-paradox-of-the-tortoise-and-achilles
     
  9. BlogPro

    BlogPro Jr. VIP Jr. VIP

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  10. davids355

    davids355 Jr. VIP Jr. VIP

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    Yup :) the great days.
    Funny though I just read the wiki link I posted and I didn't realise how influential those guys were in the hacking community.