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.
The language with ^ can definitely represent everything up to 10^280+10^273+10^266+ ... = 10000001000000100000010000001000000100000010000001000000100000010000001000000100000010000001000000100000010000001000000100000010000001000000100000010000001000000100000010000001000000100000100000100000100000100000100000100000100000100000100000100000100000100000100000100000100000999. But, I'd be surprised if that's the end of it.
Can you get f(10), where f(x) = x^3+\frac{x^{98}-x^{287}}{1-x^7}+\frac{x^8-x^{98}}{1-x^6}?
10^{280}. We can get every smaller natural by just listing the digits. Now consider an expression in the alphabet {0,...,9,+,*} (with division and subtraction removed) that isn't just a number. The expression's tree has a top level operation + or *, and both are monotone in both of their arguments. So consider a bottom-level operation a+b or a*b. Define a new operation anb which is just a string of 9's with the same length as "a+b". Since a+b<anb and a*b < anb, we can simplify the tree by replacing some bottom-level operation with n, while increasing the number represented.
The operations / and - are annoying because they aren't monotone and might leave the natural numbers. But, they can't possibly lead to getting something bigger --- a fact that deserves a proof that I will leave to the reader.
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'?
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.
Wait, both questions are pretty easy! Better throw "^" into the alphabet!
The language with ^ can definitely represent everything up to 10^280+10^273+10^266+ ... = 10000001000000100000010000001000000100000010000001000000100000010000001000000100000010000001000000100000010000001000000100000010000001000000100000010000001000000100000010000001000000100000100000100000100000100000100000100000100000100000100000100000100000100000100000100000100000999. But, I'd be surprised if that's the end of it.
Can you get f(10), where f(x) = x^3+\frac{x^{98}-x^{287}}{1-x^7}+\frac{x^8-x^{98}}{1-x^6}?
What is the smallest number not expressible in your first language?
10^{280}. We can get every smaller natural by just listing the digits. Now consider an expression in the alphabet {0,...,9,+,*} (with division and subtraction removed) that isn't just a number. The expression's tree has a top level operation + or *, and both are monotone in both of their arguments. So consider a bottom-level operation a+b or a*b. Define a new operation anb which is just a string of 9's with the same length as "a+b". Since a+b<anb and a*b < anb, we can simplify the tree by replacing some bottom-level operation with n, while increasing the number represented.
The operations / and - are annoying because they aren't monotone and might leave the natural numbers. But, they can't possibly lead to getting something bigger --- a fact that deserves a proof that I will leave to the reader.
Very nice!
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'?
I think that is just how the ultrafinitists would view it.