Welcome to the Lectures on Infinity, my series of lectures exploring all my favorite paradoxes and conundrums.
In this instance, we shall consider the paradox of the largest tweetable number. What is the largest number that you can tweet? (to use the old terminology for making a post on X) The question will lead us on a cosmic journey to some truly astounding numbers and beyond, eventually into an evaporating mist of confusion, of logical complexity and paradox, ultimately revealing a fundamental underlying issue of mathematical determinacy—does every mathematical question have a determinate answer?
A transcript of this lecture is available.
Find the lectures here on Infinitely More in the lectures-on-infinity tag.
The lectures will also appear on YouTube.
The whole lecture course is hosted at Ergo: Lectures on Infinity.
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This lecture is based on my essay The largest tweetable number.
The essay also appears in my new book, The Book of Infinity.
One way forward is to replace "tweetable" with something we can define, like: strings of length at most 280 on the alphabet {0,1,2,3,4,5,6,7,8,9,+,-,*,/} that are well-formed expressions with the usual meanings attached to the symbols. I think I could work out the largest natural number that can be so expressed, but finding the smallest number that cannot be so expressed feels rather difficult.
It strikes me that there is some point of contact between this paradox and the finitists and/or some intuitionists and/or Reeb. On the one hand, a googolplex - arguably we can all hold the concept of that - the program that emits it - in our heads. So perhaps everyone in the tent agrees that it 'exists'. On the other hand, some arbitrary number 'a bit smaller' for which we lack any concise description: we can't exhibit a program that produces it, and we arguably can't be said to hold a concept of it. So perhaps it doesn't 'exist'?