A paradox of medieval logic concerning the behaviour of an ass who is placed equidistantly from two piles of food of equal size and quality. Assuming that the behaviour of the ass is entirely rational, it has no reason to prefer one pile to the other and therefore cannot reach a decision over which pile to eat first, and so remains in its original position and starves.

Consider all English phrases that uniquely identify a real number, such as "that positive real number whose square is two", or "the ratio between a circle's diameter and its circumference. For any given positive integer n, there are a finite number of such phrases of length n (this might be zero if there are no meaningful phrases of length n, such as when n is 1 or 2). By listing first all phrases of length 3 in some order (say, alphabetical order), then all phrases of length 4, and so on, we generate an infinite ordered list of all such phrases. Call this the "Richard phrase list".

Now define a real number as follows:

"The first uninteresting positive whole number"

"This statement is false/unprovable"

How wonderful that we have met with a paradox. Now we have some hope of making progress.
- Niels Bohr

A journey of a thousand miles begins, obviously, with a single step. But isn’t it equally obvious that a step of a single metre must begin with a single millimetre? And before you can begin the last micron of that millimetre, don’t you have to get through 999 other microns first? And so ad infinitum? That “ad infinitum” bit is what worried the philosopher Zeno of Elea. Can our every action really consist of sub-actions each consisting of sub-sub-actions ... so that before we can move at all, we have to perform a literally infinite number of distinct, consecutive actions?

The problems that exist in the world today cannot be solved
by the level of thinking that created them.
- Albert Einstein

Alice laughed. "There's no use in trying," she said: "one can't believe impossible things."
"I daresay you haven't had much practice," said the Queen. "When I was your age, I always did it for half-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."
- Lewis Carroll, Alice in Wonderland

Banach-Tarski Paradox

Allais paradox

Braess�s Paradox
How the addition of an extra route can make things worse.

Braess�s paradox is an important consideration for the analysis of any network system that has alternative routes. To summarise Braess�s paradox: you sometimes increase congestion by increasing a choice of routes. This is a similar finding to the interaction-based paradox, where an increase in the number of interactions can reduce the overall system flow performance. Typically you find examples of the paradox in city traffic design and network analysis.

Place to discuss Newcomb's problem

Place to discuss Zeno's paradox

Place to discuss apparent paradoxes of quantum physics

Place to discuss apparent paradoxes of Special Relativity