Topics Mathematics

Question Suppose we decide to let 'Steve' name the successor of the largest number anyone has ever thought about before next Tuesday. Can I now think about Steve? For example can I think (or even know) that Steve is greater than 2? If not, why not? If so, wouldn't that mean that some numbers are greater than themselves?

Accepted:
October 17, 2005

Comments

It is tempting to think that the phrase "the successor of the largest number anyone has ever thought about before next Tuesday" unambiguously defines a number. After all, it seems that we could compute the value of "Steve" as follows: Wait until next Tuesday, and then make a list of all the numbers anyone has ever thought about (a finite list, given the finite history of humans thinking about numbers), find the largest number on the list, and add one.

But what counts as "thinking about a number"? From your question, it appears that you want to count thinking about a number by means of a description of that number, including descriptions like ... the definition of Steve! So the computation of the value of Steve isn't as simple as it sounds. What will happen next Tuesday when we sit down to compute the value of Steve? Well, you and I have been thinking about Steve, so when we go to make our list of all the numbers anyone has ever thought about, Steve will be on the list. This means that as part of the computation of the value of Steve, we will have to compute the value of Steve. Now it seems that the definition of Steve is circular: You can't use the definition of Steve to determine the value of Steve unless you already know the value of Steve.

Suppose I define a number "George" as follows: "George" is to be a name for the successor of George. Have I defined a number that is greater than itself? Of course not; my definition is blatantly circular, and fails to define anything. It seems to me that your definition contains a similar circularity, although the circularity is more subtle and harder to detect.

The circularity is more subtle because your definition of "Steve" doesn't refer directly to Steve. Rather, it refers to a certain collection of numbers--namely, numbers anyone has thought about before next Tuesday--and Steve belongs to that collection. Definitions like this, that don't refer directly to the object being defined, but rather refer to a collection to which the object belongs, are called impredicative, and their status in mathematics has been a subject of controversy. Your example is a nice illustration of how impredicativity can be used to make a circular definition appear not to be circular.

This question poses a version of Richard's paradox. (That's French: RiSHARD.) It's clear that not every number can be named using an expression of English that contains fewer than twenty-five syllables. There are only finitely many such expressions, after all. So there are some numbers that are not namable using fewer than twenty-five syllables, and it therefore follows from the least number principle (which is equivalent, under weak assumptions, to mathematical induction) that there is a least number that is not namable using fewer than twenty-five syllables. But now consider the phrase "the least number not namable using fewer than twenty-five syllables". It has twenty-four syllables, and so it would seem that the least number not namable using fewer than twenty-five syllables can be named using only twenty-four. Contradiction.

What Richard's paradox shows is that the notion of namability or definability needs to be treated with great care. A great deal of interesting mathematics was done in the 1930s as people struggled to work out how to do that.