Arrow's Impossibility Theorm

I seem to be posting a lot about paradoxes recently. I'll probably take a little break from paradoxes after this one.

Summary
In 1951, Kenneth Arrow demonstrated that it is not possible to have a "fair" voting system that satisfied the following three criteria (imagine the group is voting on which fruit to eat: apples or pears):

1. If every voter prefers apples to pears, then the group prefers apples to pears. (Sound familiar? It's called Pareto efficiency.)
2. If every voter prefers apples to pears, then even if bananas are added to the set of options, the group will still prefer apples to pears.
3. There is no dictator.

This is known as Arrow's Impossibility Theorem.

(Photo: Wikimedia)

Commentary
The actual details of the theorem are interesting, and I refer you to Wikipedia (for those who are interested). There are situations, however where item 2 (where we added bananas) doesn't hold: imagine the game rocks-paper-scissors. In such a case, adding an alternative transforms the straightforward choice into a cyclic choice. I sometimes see this scenario when people compare different aspects of multiple candidates' platforms (or when they're choosing which car / laptop / soap / pants to purchase).

Sometimes, the trade-offs are hard; but sometimes they're impossible.

Zeno's Paradox

Summary
Zeno's Paradox involves a race between a tortoise and Achilles that suggests that motion is an illusion.

This is not a picture of Zeno of Elea.
(Photo: Wikimedia)

Commentary
I remember when I was first introduced to this paradox in 8th grade; it was utterly puzzling (until I heard a solution). A simplified version goes something like this:

Achilles and the tortoise are having a 1000 paces race, but the tortoise has a head start of 800 paces (Achilles is much faster than the tortoise). As the tortoise inches forward, Achilles makes a plan.

First, he'll get to the halfway point (let's call it Bob) between himself and the tortoise. Of course, to get to Bob, Achilles realizes, he has to get to the halfway point between himself and Bob (let's call it Jane). And before he can consider anything else, he must first get to the halfway point between himself and Jane (called Sam). [...]

As Achilles continues to think through his plan, he realizes he can never even catch up to the tortoise, let alone win the race.

When viewed abstractly, the problem seems to show that nothing can ever pass anything else-- that is the motion is an illusion. Until the early 20th century, there wasn't really a way to handle this paradox appropriately. However, with the advent of infinite series, we can say that the reason Achilles does pass the tortoise is because if you add up all the little pieces (1/2 + 1/4 + 1/8 + 1/16 ...) you get 1 (which represents the total distance between Achilles and the tortoise).

Fun fact: Zeno's paradoxes are considered some of the earliest examples of reductio ad absurdum, also known as proof by contradiction.

Russell's Paradox

Summary
Russell's paradox describes a fundemental conundrum with set theory that is sometimes illustrated by way of a story about a barber.

(Photo: Wikimedia)

Paradox
This presentation is a modified version of Russell's original presentation.

There once was a platoon of men who were very punctilious about following orders. One day, the commander decided that the men needed to look less disheveled. One of the men, a barber named Bob, proposed that he shave everyone everyday. Some of the men complained and asked if they could shave themselves instead. As a compromise, the commander ordered Bob to shave all and only the men who do not shave themselves.

The paradox is this: who shaves Bob? If he shaves himself, then he ends up being prohibited from shaving himself. In which case he must shave himself. Continue ad infinitum.

Commentary
The significance of this paradox was that it undermined the existence of certain types of sets, potentially undermining all of set theory. There were some interesting responses, but ultimately Gödel showed the inescapable nature of such paradoxes for most logical systems.