Posted by: dschneid2010 | October 10, 2008

I really like paradoxes. I took a class that involved set theory, and, very soon into the class, you begin to see stuff that seems “just not right”, or at least, not intuitive. I went shopping around the internet to see what cool paradoxes I could find (some set theory, but not all). Here’s a few of many.

Essentially, Russel’s paradox is the following: Imagine a set that contains all sets that do not contain themselves. Does this set contain itself?

Well, if it doesn’t contain itself, then, the set in question, does not contain itself, and should be in itself as a set that does not contain itself…but then it would be in it self…

There’s a lot of variations of this that put it into less confusing English, for example: A barber shaves everyone in town who does not shave themselves. Does the barber shave himself?

Or

Imagine a book of title’s of books that do not contain their own title. Is the title of this book in itself?

I think a whole revamping of set theory was done after the discovery of this paradox.

Fill a huge reservoir with balls enumerated by numbers 1 to 10 and take off ball number 1. Then add the balls enumerated by numbers 11 to 20 and take off number 2. Continue to add balls enumerated by numbers 10n – 9 to 10n and to remove ball number n for all natural numbers n = 3, 4, 5, …. Let the first transaction last half an hour, let the second transaction last quarter an hour, and so on, such that all transactions are finished after one hour. Obviously the set of balls in the reservoir increases without bound. Nevertheless, after one hour the reservoir is empty because for every ball the time of removal is known.

Consider the following scenario: Two closed boxes, B1 and B2, are on a table. B1 contains \$1,000. B2 contains either nothing or \$1 million (you do not know which). You may choose either to (a) take the contents of both boxes, or (b) take only what is in B2. Some time before the test, an entity who is able to make highly accurate predictions about your decisions has made a prediction about what you will decide. If the entity expects you to choose both boxes (or expects you to randomize your choice), he has left box B2 empty. If he expects you to take only B2, he has put \$1 million in it. The paradox lies in the fact that there are valid reasons for choosing either (a) or (b). If you take both boxes, the entity will almost certainly have anticipated that and left B2 empty, whereas if you take only B2, the entity will almost certainly have anticipated that, and put \$1 million in B2, so you should take B2. However, either the money is already in B2 or it isn’t. It will stay there whatever you choose. So, whether there is \$1 million in B2 or not, you will always make \$1,000 more by choosing both boxes.

Consider a game in which a coin is flipped until heads comes up. If it comes up on the first toss, you win \$1. If it comes up on the second toss, you win \$2. If it comes up on the third toss, you win \$4. In general, if it comes up on the nth toss, you win \$2 n-1. The expectation (average amount you can expect to win) for this game can be found by adding together the products of the probabilities for each outcome with the amount that is won for each outcome. This gives us (1/2) x 1 + (1/4) x 2 + (1/8) x 4 + (1/16) x 8 + . . . = ½ + ½ + ½ + ½ + . . .

Since the number of terms in this sequence is infinite, the sum is also infinite. Therefore, the expectation for this game is infinite. The paradox lies in the fact that it doesn’t make sense to have an infinite expectation, since it is only possible to win finite amounts of money.

There are as many square numbers as there are integers and vice versa. This is exhibited in the correspondence
1 <–> 1
2 <–> 4
3 <–> 9
4 <–> 16
5 <–> 25
. . .
But how is this possible when not every number is square? The answer is that both sets are
infinte sets with the same cardinality.

Hope no one’s world has been turned upside down!

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