Paradoxes

A paradox is a statement that goes against our intuition but may be true, or a statement that is self-contradictory. The paradoxes listed below and most other mathematical paradoxes fall into one of two categories: either they result from the counter-intuitive properties of infinity, or are a result of self-reference.

Some Famous Paradoxes:

Zeno's Paradox

Russell's Paradox

One can classify sets into one of two categories. The first category is sets that are not members of themselves. This contains most of the sets we run into in "real life". For example, the set of all penguins falls in the first category, because the set of all penguins is a set, not a penguin. On the other hand, some sets are members of themselves. The set of all non-penguins, for example, is a member of itself. So is the set of all sets.
In which category would we find the set of all sets that are not members of themselves? If this set is not a member of itself, then it is a member of itself. If it is, then it isn't. So, this set is a member of itself if and only if it is not a member of itself, which is the paradox. This is similar in concept to the Cretan Liar paradox.
An article about Russell's Paradox at the Stanford Encyclopedia of Philosophy.

Greeling's Paradox

A version of Russell's Paradox using words. Some adjectives are self-descriptive, like "tiny", "unhyphenated", and "pentasyllabic". On the other hand, other adjectives are not self-descriptive, like "bisyllabic", "big", "tasty", and "incomplete". Call the self-descriptive adjectives autological, and the non-self-descriptive adjectives heterological. Now, is "heterological" autological or heterological? If it is, then it isn't. If it isn't, then it is. Either way, there's a paradox.

Barber shop paradox

A version of Russell's paradox. In the town barber shop, the (male) barber puts a sign up which states that he shaves all men in the town who don't shave themselves, and only those men. Does the barber shave himself or not?

Cretan Liar paradox

Another version of Russell's paradox. One variant of this is as follows: Someone states: "This statement is false". Is it false or not? If it's false, then the statement is true. If it's true, it's false.

Thomson Lamp paradox

We have a perfect machine for turning a light on and off. First, we have the light on for one minute, after which it is turned off for one half of a minute. Then it is on again for one fourth of a minute and off for one eighth of a minute. This continues with the light turned on or off after one half of the preceding time period. After two full minutes an infinite sequence of offs and ons will have occured. At this time, will the light be on or off?

G.G. Berry's Paradox

We can classify the integers based on the smallest number of syllables in English necessary to describe them. Consider the set of all integers that require at least nineteen syllables to describe them. This set will have a smallest element. However, we can describe this integer as the "least integer not describable using less than nineteen syllables", which is a description of eighteen syllables! Therefore, there is no least integer that requires nineteen syllables to be described.

Galileo's Paradox

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 infinite sets with the same cardinality.

Rearrangement paradoxes

These paradoxes say that the sum of an infinite series may be changed by rearranging its terms. For example, 0 = (1 - 1) + (1 - 1) + (1 - 1) + . . . = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + . . . = 1 + 0 + 0 + 0 + . . . = 1. Therefore, 0 = 1.

Newcomb's Paradox

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.

In game theory, we might say that there is a conflict between the "expected-utility principle" and the "dominance principle". Assuming that the being is able to predict with near certainty, the expected utility of taking only box B2 is much larger (you're very likely to take $1 million, whereas if you take both boxes, you're very likely to only get $1000). On the other hand, no matter what is in the boxes, the option to take both boxes dominates the option to only take box B2.

Petersburg Paradox

This is a paradox in the field of game theory. 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 n^th 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.