Vicious Circle Paradoxes

Vicious circle paradoxes are types of paradoxes that arise from a certain kind of self-reference. Here are some examples of these:

Russell's Paradox

Many sets are not members of themselves. For example, the set of all elephants is a set, not an elephant. The set of all odd numbers is not an odd number. In contrast, it is possible to conceive of a set that is a member of itself; for example, the set of all sets is a set, and so would be a member of itself. Now, consider the set of all sets that are not members of themselves. Is this set a member of itself? If it is, then it isn't. If it isn't, then it is.

Barber Paradox

In a town, there is a single barber shop. The (male) barber claims that he shaves all men in the town who don't shave themselves, and only those men. Now, does the barber shave himself? If he does, then by his rule he doesn't, and if he doesn't, then he does. While not strictly a logical paradox (it's not logically impossible for the barber to ignore his own rule), it can help to make Russell's paradox easier to understand.

Cantor's Paradox

The cardinal number of any set is less than the cardinal number of the set of all subsets of that set. For example, the cardinal number of {1, 2} is 2; the set of all possible subsets of {1, 2} is {{}, {1}, {2}, {1, 2}}, and its cardinal number is 4. Now, consider the set of all sets, U. Its cardinal number must be less than the set of all possible subsets of U; however, the set of all possible subsets of U must be a subset of U as its members are all sets, so U must have a greater cardinal number.

Cretan Liar Paradox (also known as the Epimenides Paradox or the Liar Paradox)

Epimenides of Crete is said to have said that Cretans are always liars. If they're always liars, then, since Epimenides is from Crete, his statement must also be a lie, so Cretans aren't always liars and around we go in a circle again. There are many other ways of expressing this paradox; perhaps the simplest is that of someone who says, "Everything I say is false," or "This statement is false."

Richard's Paradox

Let E be the set of all non-terminating decimals that can be defined in a finite number of words. Since the definitions can be arranged in a sequence in, say, alphabetical order, E can also be arranged in a sequence. Now, form a new non-terminating decimal as follows: Let the nth digit of N be 1 greater than the nth digit of the nth member of E, or 0, if that digit is 9. Now, N cannot be the same as any of the members of E, because it differs in at least one digit with each of them. However, we have just defined N in a finite number of words.

G. G. Berry's Paradox

We could classify the positive integers in terms of the smallest number of syllables required to describe them in English. For example, 1 could be described in one syllable, "one." We could get creative and describe 1,728 in two syllables as "twelve cubed," and the like. No matter how creative we get, though, there are only a finite number of words and syllables in English, so, for any given number of syllables, say 18, there will be only a finite number of natural numbers that can be described with no more than that number of syllables. Since the natural numbers are infinite, some numbers cannot be described using 18 or fewer syllables. Since the natural numbers are well-ordered, there must be a smallest such integer. This integer can be described "the least integer not describable in less than nineteen syllables," which describes the integer using only eighteen syllables. But this integer can't be described using eighteen syllables, so we have a contradiction.

Grelling's Paradox (also known as the Grelling-Nelson Paradox)

Some adjectives, such as "English" or "tiny," apply to themselves. Others, such as "German" or "long," do not. Let the latter type of adjectives be described as heterological. Now, is "heterological" heterological or not? Whether it is or not, a contradiction results.

These paradoxes all share in common the fact that they result from situations where a set or some other collection contains members that are defined via the collection as a whole. For example, if someone says, "Everything I say is false," then "everything" would include the sentence "Everything I say is false," resulting in a vicious circle.

These paradoxes are rather amusing, but they also are significant. For example, Russell's paradox revealed a contradiction in the set theory current at the time, a big problem since set theory is one of the foundations of mathematics. Various ways to resolve this paradox were found in order to make set theory consistent again. Russell proposed the "vicious circle principle," which states that no member of a collection can be defined in terms of the collection itself. Others developed new axioms for a new set theory that would avoid these paradoxes. This set theory would become what is now called Zermelo-Fraenkel set theory.