You might wonder why we can’t transcend the whole parade of paradigms, and name numbers by a system that encompasses and surpasses them all. Suppose you wrote the following in the biggest number contest:
The biggest whole number nameable with 1,000 characters of English text
Surely this number exists. Using 1,000 characters, we can name only finitely many numbers, and among these numbers there has to be a biggest. And yet we’ve made no reference to how the number’s named. The English text could invoke Ackermann numbers, or Busy Beavers, or higher-level Busy Beavers, or even some yet more sweeping concept that nobody’s thought of yet. So unless our opponent uses the same ploy, we’ve got him licked. What a brilliant idea! Why didn’t we think of this earlier?
Unfortunately it doesn’t work. We might as well have written
One plus the biggest whole number nameable with 1,000 characters of English text
This number takes at least 1,001 characters to name. Yet we’ve just named it with only 80 characters! Like a snake that swallows itself whole, our colossal number dissolves in a tumult of contradiction. What gives? ... but from the ambiguity inherent in the English language. There’s no surefire way to convert an English phrase into the number it names
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