It is a fine thing to have a paradox to ponder. This is one that I heard as gossip when I was on the math competition circuit in my high school days.
For today, I mean “number” in the sense of natural number. 0, 1, 2, 3, and so on. All the things that might be the answer to a “How many” question.
Consider all the ways to name numbers with English words. So there’s “four”, of course, for 4, but there’s also “seven minus three” and “two squared” and “the smallest number that is not a factor of thirty” and lots of other things.
Maybe we even accept things like “the number of legs on a healthy dog” or “the number of sides that a trapezoid has”. The most important things are that names consist of strings of English letters and spaces, and must unambiguously refer to a single natural number. We could make various decisions about what kinds of operations are allowed, and what context is assumed, whether time-specific things such as “the number of letters in the middle name of the president” count, etc. The exact details of the conventions we take here are not important, so let’s just suppose we’ve laid down what it means to have a “name” of a number. A number can have many, many names, and a string of words may or may not actually be the name of any number.
Now let’s think about what happens if we limit the length of our names, say, to 60 letters maximum. Then the total number of combinations of at most 60 letters from the English alphabet is a finite (enormous) number. Even if you factor in where the spaces go, it is still finite. You could make a list of all the strings of spaces and letters containing at most 60 letters and no two consecutive spaces. (It would be a long list, but a finite one.) Now, the vast majority of these strings are unpronounceable garbage like “akljta df grt ar sfg rt dfg”. Of those consisting of bona fide words, the vast majority are nonsense like “whereupon cat clavichord”. Of those that are discernibly a noun phrases, the vast majority do not name a number, like “a cloudy day”. Of those that do name numbers, the many are duplicates. When all is said and done we have only found names for finitely many different numbers. If you had time enough, you could actually figure out which numbers they are, and you could figure out which is the first number not among them. In other words, you could, in principle, identify the first number which cannot be named with sixty letters or fewer. What an interesting number that must be, huh?
But then “the smallest number that cannot be named with sixty letters or fewer” is a name for that very number, and we have done it with just 57 letters.
(There seems to be some ambiguity over who first suggested this paradox; I’ll shout if I can find something definitive.)