Richard's Paradox

One form of a paradox is a sentence that appears to contradict itself.  For example:

“This sentence is false”

If the sentence is true, then it has to be false (because that is what it says).

If the sentence is false, then it has to be true (because is it false that “This sentence is false.”).

It can also apply to mathematical statements, such as Richards Paradox.

Clearly there are integers so huge they can’t be described in fewer than 22 syllables. Put them all in a big pile and consider the smallest one. It’s “the smallest integer that can’t be described in fewer than 22 syllables.”  But, that phrase has 21 syllables.

Remember Richard’s Paradox – we will be using it again soon.

David