This one is cool!

http://en.wikipedia.org/wiki/Richard's_paradox

The paradox begins with the observation that certain expressions in English unambiguously define real numbers, while other expressions in English do not. For example, "The real number whose integer part is 17 and whose

*n*th decimal place is 0 if

*n*is even and 1 if

*n*is odd" defines the real number 17.1010101..., while the phrase "London is in England" does not define a real number.

Thus there is an infinite list of English phrases (where each phrase is of finite length, but lengths vary in the list) that unambiguously define real numbers; arrange this list by length and thendictionary order, so that the ordering is canonical. This yields an infinite list of the corresponding real numbers:

*r*

_{1},

*r*

_{2}, ... . Now define a new real number

*r*as follows. The integer part of

*r*is 0, the

*n*th decimal place of

*r*is 1 if the

*n*th decimal place of

*r*

_{n}is not 1, and the

*n*th decimal place of

*r*is 2 if the

*n*th decimal place of

*r*

_{n}is 1.

The preceding two paragraphs are an expression in English which unambiguously defines a real number

*r*. Thus

*r*must be one of the numbers

*r*

_{n}. However,

*r*was constructed so that it cannot equal any of the

*r*

_{n}. This is the paradoxical contradiction.