# PARADOXES

Achilles
and the Tortoise paradox

Allais paradox

Aristotle's
wheel

Arrow paradox

Banach-Tarski
paradox

Berry's paradox

Bertrand's
box paradox

birthday
paradox

Burali-Forti
paradox

Buridan's ass

## Catch-22

A Catch-22 is a situation in which a person is frustrated by a paradoxical rule or set of circumstances that preclude any attempt to escape from them. The name comes from the novel by Joseph Heller (1923–1999), based on his personal experiences, about an American airman's attempts to survive the madness of the Second World War. Heller wrote: "There was only one catch and that was Catch-22, which specified that concern for one's own safety in the face of dangers that were real and immediate was the process of a rational mind. Orr was crazy and could be grounded. All he had to do was ask; and as soon as he did, he would no longer be crazy and would have to fly more missions. Orr would be crazy to fly more missions and sane if he didn't, but if he was sane he had to fly them. If he flew them he was crazy and didn't have to; but if he didn't want to he was sane and had to."

## coin paradox

Consider two round coins of equal size. Imagine holding one still and then
rolling the other coin around it, making sure that it doesn't slip and that
the rims are touching at all times. How many times will the moving coin
have rotated after it has completed one revolution of the stationary coin?
Most people believe that the answer will be once and are therefore surprised
to discover that the truth is in fact *twice*.

Epimedes paradox

## Grelling's paradox

Grelling's paradox is an equivalent, from the world of words and grammar, of Russell's
paradox. It involves dividing all adjectives into two
sets: self-applicable and not self-applicable. Words like "English," "written,"
and "short" are self-applicable, while "Russian," "spoken," and "long,"
are not self-applicable. Now, define the adjective *heterological* to mean "not self-applicable." To which set of adjectives does "heterological"
belong? This strange quandary was devised by the logician and philosopher
Kurt Grelling (1886–1941/2), who was persecuted by the Nazis; it is
not certain whether he died with his wife in the Auschwitz concentration
camp in 1942, or whether he was killed in 1941 in the Pyrenees while trying
to escape into Spain.

liar paradox

Newcomb's paradox

nine rooms paradox

## paradox

Please accept my resignation. I don't want to belong to any club that will accept me as a member. |

—Groucho Marx (1895-1977) |

A paradox is a statement that seems to lead to a logical self-contradiction, or to a
situation that contradicts common intuition. The word "paradox" comes from
the Greek *para* ("beyond") and *doxa* ("opinion" or "belief").
The identification of a paradox based on seemingly simple and reasonable
concepts has often led to significant advances in science, philosophy, and
mathematics.

## Parrondo's paradox

Two losing gambling games can be set up so that when they are played one after the other, they become winning. This paradox is named after the Spanish physicist Juan Parrondo who discovered how to construct such a scenario.

The simplest way is to use three biased coins. Imagine you are standing on stair zero, in the middle of a long staircase with 1001 stairs numbered from -500 to 500. You win if you can get to the top of the staircase, and the way you move depends on the outcome of flipping one of two coins. Heads you move up a stair, tails you move down a stair. In game 1, you use coin A, which is slightly biased and comes up heads 49.5% of the time and tails 50.5%. Obviously, these are losing odds. In game 2, you use two coins, B and C. Coin B comes up heads only 9.5% of the time, tails 90.5%. Coin C comes up heads 74.5% of the time, tails 25.5%. In game 2 if the number of the stair you are on at the time is a multiple of 3 (that is, ..., –9, –6, –3, 0, 3, 6, 9, 12, ...), then you flip coin B; otherwise you flip coin C. Game 2, it turns out, is also a losing game and would eventually take you to the bottom of the stairs. What Parrondo found, however, is that if you play these two games in succession in random order, keeping your place on the staircase as you switch between games, you will steadily rise to the top of the staircase!

### Reference

1. Harmer, G. P. and Abbott, D. "Losing Strategies Can Win by Parrondo's
Paradox." *Nature*, 402: 864 (1999).

raven paradox

St. Petersburg paradox

## Siegel's paradox

Siegel's paradox is a way of investing in foreign investments to make money. If a fixed fraction *x* of a given amount of money *P* is
lost, and then the same fraction *x* of the remaining amount is gained,
the result is less than the original and equal to the final amount if a
fraction *x* is first gained, then lost.

twins paradox

unexpected hanging

Zeno's paradoxes