The Limitations of Science Trivia Quiz

Answer: Bertrand Russell

Let S be the set of all sets which do not contain themselves. Does S contain itself? Either possibility leads to a contradiction!

What this means is that we cannot develop a perfectly general set theory. While we would like to be able to just take all sets which satisfy a certain condition and call that a set, we can't always do this without possibly introducting a contradiction. In order to get around this Russell and his teacher Whitehead developed a theory of "types" (basically an infinite hierarchy of sentences). This theory was described in ever more obscure volumes (most notably "Principia Mathematica") in an attempt to avoid this paradox and develop a complete and consistent set theory. However...(see question 2)

Answer: Kurt Gödel

"Sufficiently complex" in this context isn't really that complex at all -- it only means that the system can do arithmetic! As an example of a true but unprovable statement in English, consider the sentence "This sentence cannot be proven true." If it is false, you certainly can't prove it true, so we have a contradiction. Therefore it must be true, but by definition then it is unprovable.

That such a sentence exists in English (which is a very vague language) is not surprising. Gödel showed that a similar sentence existed for every logical system created by mathematicians, no matter how intricately designed. Even if you assume as an axiom that some unprovable statemtent is true (thus eliminating the need to prove it), this will only cause more unprovable statements to be created. This problem occurs even in the system of symbols Russell and Whitehead spent years designing in an attempt to get around Russell's Paradox!

Answer: David Hilbert

For any specific Diophantine equation, we may be able to come up with a way of figuring out if there are solutions (though this may be difficult; Fermat's Last Theorem took hundreds of years!). Yuri Matijasevich showed (basing his work upon earlier work of Julia Robinson) that no one method could work for ALL equations.

Answer: The Axiom of Choice

In some cases, this can be done without a special axiom (if you had infinitely many pairs of shoes, for example, simply take all the left shoes). The axiom states that it can always be done regardless of the set in question.

It is called "unanswerable" because you can create a perfectly consistent system of arithmetic where the axiom is true, and another consistent system where it is false. Because of this, mathematicians tend to try to avoid using this axiom when proving things. In some cases, however, avoidance of the axiom is impossible.

Answer: The speed of light in a vacuum

Relativistic formulae for the mass, length, and time dilation of an object (among others) contain a factor of 1 over the square root of (1-v' squared), where v' is the ratio of an object's speed to that of light in a vacuum. If v' was allowed to reach 1, we would be dividing by 0 and the bad consequences would pile up pretty quickly.

Objects CAN in fact travel faster through a medium than light travelling through the same medium. When they do, they emit the visual equivalent of a sonic boom in a process known as Cherenkov radiation.

Answer: Werner Heisenberg

In Quantum Mechanics, particles' locations are typically expressed as probability wavefunctions over space. Instead of saying that a particle is located precisely here, we would say a particle is located in a certain region with a certain probability.

What Heisenberg's theorem actually says is that the uncertainty in position (defined as the standard deviation over many measurements of the particle's position) times the uncertainty in momentum (defined similarly) is at least a nonzero constant. A similar theorem holds for any two operators which do not commute with each other (energy and time, for example).

Answer: Kenneth Arrow

The conditions Arrow imposed on his election were as follows. First, that every possible set of votes had to lead to some result. Second, that every possible ranking of candidates was in fact realizable in some manner (i.e. any candidate could conceivably win the election). Third, that no candidate be penalized for receiving extra votes. Fourth, that a candidate that stood no chance of winning would not effect the ranking of the other candidates (think Gore/Nader or Bush/Perot).

This served as a rather severe blow for election theorists, many of whom had attempted to design more and more complex systems in order to get around "flaws" they saw in various systems (especially Arrow's third and fourth conditions). Arrow basically said that all those theorist's efforts were hopeless -- NO system could avoid one of those flaws.

Answer: Yes

If the other prisoner speaks, speaking out will give you 3 years in prison as opposed to 5. If he is silent, you will go free instead of spending a year in prison. Note the end result here: Each prisoner "should" speak out and get 3 years in prison, but if they both had only stayed silent, they could have gotten off with only a year!

Ever since Adam Smith and his "invisible hand," many economists had assumed that a market will reach its optimal point simply by letting each individual act freely and rationally. This example shows that the theory of "Laissez-Faire" (letting alone) is not sufficient in all situations. More complex theories of welfare economics are in fact necessary.

Answer: Alan Turing

Turing used a cute trick to prove this. Suppose we had a program "Awesome" that returned "yes" to all programs that stopped in finite time and "no" to programs that didn't. Consider a program "Trouble" that enters an infinite loop if "Awesome" returns "yes" and stops otherwise. What would happen if we inputted this program into "Awesome"? If "Awesome" claims that "Trouble" halts, "Trouble" enters an infinite loop, while if "Awesome" claims that "Trouble" doesn't halt, "Trouble" stops! This is a contradiction, so no such program "Awesome" exists!

Answer: Claude Shannon

Depending on the frequency of certain symbols, we can often communicate the same message in less space by developing some sort of coding algorithm. For example, suppose we replaced all the q's in the english language by th and all the th's by q. We could still decode the message by simply swapping the q's and th's back. The message will be shorter on average, however, since th's occur more frequently than q's.

While such compression is desirable, Shannon proved that we can only compress data so much. Any method which compresses data beyond Shannon's bounds (its so-called "entropy") must not be entirely decodable. This is why MP3 and similar audio files on your computer often don't sound as good as the original CD. The "desirable" size for such a file is far below the Shannon bound, so to reach that size information HAS to be lost!