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.
In 2007 two philosophy professors — Adam Elga and Agustin Rayo — 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.
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.
The number picked out by Rayo’s description has come to be called, appropriately enough, Rayo’s number.
Can we come up with short descriptions of even bigger numbers?
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).
Source: Really big numbers
Originally published at Cogly.
