What is the biggest whole number that you can write down or describe uniquely? Well, there isn’t one, if we allow ourselves to idealize a bit. Just write down “1”, then “2”, then… you’ll never find a last one.

Of course, in real life you’ll die before you get to any really big numbers that way. So here’s a more interesting way of asking the question: what is the biggest whole number that you can uniquely describe on a standard sheet of paper (single spaced, 12 point type, etc.) or, more fitting, perhaps, in a single blog post?

In 2007 two philosophy professors – Adam Elga (Princeton) and Agustin Rayo (MIT) – asked essentially this question when they competed against each other in the Big Number Duel. The contest consisted of Elga and Rayo taking turns describing a whole number, where each number had to be larger than the number described previously. There were three additional rules:

Any unusual notation had to be explained. No primitive semantic vocabulary was allowed (i.e. “the smallest number not mentioned up to now.”) Each new answer had to involve some new notion – it couldn’t be reachable in principle using methods that appeared in previous answers (hence after the second turn you can’t just add 1 to the previous answer)

Elga began with “1”, Rayo countered with a string of “1”s, Elga then erased bits of some of those “1”s to turn them into factorials, and they raced off into land of large whole numbers. Rayo eventually won with this description:

The least number that cannot be uniquely described by an expression of first-order set theory that contains no more than a googol (10100) symbols.

A more detailed description of the Duel, along with some technical details about Rayo’s description, can be found here.

Fans of paradox will recognize that Rayo’s winning move was inspired by the Berry paradox:

The least number that cannot be described in less than twenty syllables.

This expression leads to paradox since it seems to name the least number that cannot be described in less than twenty syllables, and to do so using less than twenty syllables! Rayo’s description, however, is not paradoxical, since although it uses far fewer than a googol symbols to describe the number in English, this doesn’t contradict the fact that, in the expressively much less efficient language of set theory, the number cannot be described in fewer than a googol symbols.

The number picked out by Rayo’s description has come to be called, appropriately enough, Rayo’s number. And it is big – really big. But can we come up with short descriptions of even bigger numbers?

Notice that Rayo’s construction implicitly provides us with a description of a function:

F(n) = The least number that cannot be uniquely described by an expression of first-order set theory that contains no more than n symbols.

Rayo’s number is then just F(10100). So one way to answer the question would be to construct a function G(n) such that G(n) grows more quickly than F(n). Here’s one way to do it.

First, we’ll define a two place function H(m, n) as follows. We’ll just let H(0, 0) be 0. Now:

H(0, n) = The least number that cannot be uniquely described by an expression of first-order set theory that contains no more than n symbols.

So H(0, n) is just the Rayo function, and H(0, 10100) is Rayo’s number. But now we let:

H(m, n) = The least number that cannot be uniquely described by an expression of first-order set theory supplemented with constant symbols for:

H(m-1, n), H(m-2, n),… H(1, n), H(0, n)

that contains no more than n symbols.

In other words, H(1, 10100) is the least number that cannot be described in first-order set theory supplemented with a constant symbol that picks out Rayo’s number. Note that, in this new theory, Rayo’s number can now be described very briefly, in terms of this new constant! So H(1, 10100) will be much larger than Rayo’s number.

But then we can consider H(2, 10100), which is the least the least number that cannot be described in first-order set theory supplemented with a constant symbol that picks out Rayo’s number and a second constant symbol that picks out H(1, 10100). This number is much, much bigger than H(1, 10100)!

And then we have H(3, 10100), which is the least the least number that cannot be described in first-order set theory supplemented with a constant symbol that picks out H(0, 10100), a second constant symbol that picks out H(1, 10100) and a third constant symbol that picks out H(2, 10100). This number is much, much, much bigger than H(2, 10100)!

And so on…

We can now get our quickly growing unary function G(n) by just identifying m and n:

G(n) = H(n, n).

And finally, our big, huge, enormous, number is:

G(10100)

G(10100) is the least number that cannot be described in first-order set theory supplemented with googol-many constant symbols – one for each of H(0, 10100), H(1, 10100), … H(10100-1, 10100).

This number really is big. Can you come up with a bigger one?

Featured image: “Infant Stars in Orion” Public domain via Wikimedia Commons.