Berry's Paradox goes something like this:
The smallest positive integer that can't be described in fewer than fifteen English words.
That got me thinking about ways to define large numbers in fewer than fifteen English words. I wanted to follow these rules:
- Well defined, no paradoxes ("Ten times ten" is allowed but "The smallest positive integer that can't be described in fewer than fifteen English words" or "Zero divided by zero" is not)
- Must be fewer than fifteen English words
- Any words used in your phrase without being defined elsewhere in your phrase must be officially defined (Google Dictionary) so if you wrote "Rayo's function of one million" you would need to use the remaining words to define Rayo's function.
- Use as many punctuation marks as you want but only for punctuation and not for functions (e.g. ! is not allowed for factorial or else you would be allowed to put "Ten" followed by arbitrarily many factorials)
- You can name your number in your phrase but you don't have to ("Googol is ten to the power of one hundred" and "Ten to the power of one hundred" are both allowed)
- You can use letters for variables but each letter counts as one word
- No digits/numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9)
- You can use contractions and they count as one word
Helpful tips:
- The largest number which is properly defined in Google Dictionary is "Googolplex"
- The fastest growing function which is properly defined in Google Dictionary is "Factorial"
The largest number I could define in fewer than fifteen English words is this one:
Googolplex “Googolplex before (”s before “Googolplex” before Googolplex “) 'factoriterated's”'s; x factoriterated's x "factorial"s after x
This piece of text defines a new function "factoriterated" (just iterated factorials) after defining how to construct a piece of text which defines how to construct a piece of text which defines this number. I put a lot of punctuation marks and contracted "x factoriterated is" to "x factoriterated's", in addition to using curly quotes for clarity.
I can also mathematically recursively define this number (I'll call it "Googolplactoriteriteritex"):
f1(x,0) = x
f1(x,y+1) = f1(x,y)!
f2(x,0) = x
f2(x,y+1) = f1(f2(x,y),f2(x,y))
f3(x,0) = x
f3(x,y+1) = f2(f3(x,y),f3(x,y))
Googolplactoriteriteritex = f3(10^(10^100),10^(10^100))
See if you can define a number bigger than this number following the above rules.