10 Favorite Paradoxes

Curry's Paradox – Proof that Santa Claus Exists

The paradox itself is too mathematical for this article but it can be described in simple language as follows:

 If everything in this sentence is true, then Santa Claus exists.

Suppose everything in the sentence is true. The sentence says that if everything in it is true, then Santa Claus exists. Since we are supposing that everything in the sentence is true, therefore we can conclude that Santa Claus exists.

We have, then, asserted that we can prove the existence of Santa Claus from the assumption that everything in the sentence is true. This is precisely what the sentence asserts. So everything in the sentence is, in fact, true. It follows therefore that Santa Claus exists!

Paradox of the Heap

1.    Two or three grains of sand do not make a heap.
2.    A million grains do make a heap.
3.    If n grains of sand do not make a heap, neither do (n+1) grains.
4.    If n grains of sand make a heap, so do (n−1) grains.

The paradox is that, contrary to (3), we might add grain after grain to our collection of sand until it truly does become a heap. And according to (4), if we began with a heap, and took single grain after single grain away from the collection, it would never stop being a heap; even if there were no grains of sand left at all.

So common language suggests that heaps of sand have the properties described in (3) and (4), since the difference of one grain of sand is thought to be negligible. But it seems that these properties are actually mutually inconsistent. That is the paradox.

Say that you are in a room with 23 people. What do you think are the chances that at least two of them will have the same birthday date?

Surprisingly, the chances are more than 50%.

Of you ask 60 random people for their birthday date. Chances are more than 99% that at least two will have the same date.

Explanation: Think of it this way:
There are 365 days a year. Imagine a yearly calendar with 365 squares.
Ask one person what is his birthday day and cover this square.
Ask the next person and the number of options that his birth date fall on one we covered is 1:364.
The 3rd person the chances are 2:363
4th – 3:362
20th 20:344
23rd 23:342

Now add the chances we had to hit a marked date throughout the process and you get:
1/364+2/363 + …+ 23/342 = ~1/2

To see a simulation of this paradox visit http://www-stat.stanford.edu/~susan/surp rise/Birthday.html

Liar paradox

The old version goes "A man says that he is lying. Is what he says true or false?"

A modern version of the Liar paradox is the two sentences:
•    I am lying now.
•    This statement is false.

There's also a shorter version: (1) This sentence is false.

Explanation: (of the shorter version)

If (1) is true, then (1) is false. On the other hand, assume (1) is false. Because the Liar Sentence is saying precisely that (namely that it is false), the Liar Sentence is true, so (1) is true. We've now shown that (1) is true if and only if it is false. Since (1) is one or the other, it is both.
(explanation from the Internet Encyclopedia of Philosophy)

Grandfather Paradox (Time travel Paradox)

The grandfather paradox was conceived by René Barjavel in his book "Future times three", 1943. It proves that time travel is not possible.

Suppose you traveled back in time and killed your biological grandfather before he met your grandmother. Then you would never have been conceived, so you could not have traveled back in time after all. In that case, your grandfather would still be alive and you would have been conceived, allowing you to travel back in time and kill your grandfather, and so on.

Twins Paradox (Clock paradox)

One of two twin brothers undertakes a long space journey with a high-speed rocket at almost the speed of light, while the other twin remains on Earth. When the traveler returns to Earth, he is younger than the twin who stayed put.

Or, as first stated by Albert Einstein (1911): If we placed a living organism in a box ... one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations. For the moving organism the lengthy time of the journey was a mere instant, provided the motion took place with approximately the speed of light. (in Resnick and Halliday, 1992)

Explanation: According to the theory of relativity a moving clock is found to experience a reduced amount of proper time as determined by clocks synchronized with a stationary clock. This phenomenon is called time dilation. In other words: Do not try this at home.

Omnipotence paradox

Is it possible that God or other creature with unlimited power and capabilities can exist?
The Omnipotence Paradox attempts to prove that such a being cannot exist.
The paradox: "Could an omnipotent being create a rock so heavy that even that being could not lift it?"

Explanation: If an omnipotence creature could exist, it would have unlimited power and capabilities. Therefore it would have power to limit its own omnipotence. If the being's omnipotence can be limited, then it is not omnipotent.

Unexpected Hanging Paradox

A judge makes two statements to a condemned prisoner:
•    You will be hanged at noon one day next week, Monday through Friday.
•    The execution will be a surprise to you: you won't know the day of the hanging until the executioner knocks on your cell door at noon that day.

The Hanging paradox proves that the hanging will not take place.

Explanation: If the hanging is on Friday than the statements fail because the prisoner will know it's going to happen since it's the last day.
If the hanging is on Thursday than the statement also fails because on Thursday morning the prisoner will know it's going to happen since it cannot happen on Friday.
The same logic rules out all weekdays.

Unexpected Hanging Paradox V.2

A judge makes two statements to a condemned prisoner:
•    You will be hanged at noon next week on Friday.
•    The execution will be a surprise to you.
The prisoner is happy because the statement is logically invalid. Since the judge told him the time of hanging so it's no longer a surprise so he cannot be hanged without breaking the judge decision.
The prisoner is very surprised when they hang him on Friday.

Fermi paradox

Explanation: The Paradox describes that in spite of the fact that the enormous size of our galaxy and others makes it statistically feasible that intelligent technological civilizations are not rare, all the methods we use searching for such civilizations failed and we cannot find none.

http://www.sff.net/people/Geoffrey.Landi s/percolation.htp

Sources: www.Wikipedia.org ,Answers.com, http://mathworld.wolfram.com/