User blog comment:Kyodaisuu/Pseudo Rayo's number/@comment-119611-20150102151948

A number can be considered ill-defined if the definition of it does not actually define a specific number. "Largest positive integer" is ill-defined, as is "smallest integer equal to its successor", or "largest odd number divisible by 4".

If you define a function f(x) as f(1)=2 and f(x)=f(f(x-1))+1 for x>1, then f(2) is ill-defined, because the definition says f(2)=f(f(1))+1=f(2)+1. If you say f(x)=f(f(x-1)) instead, then f(2) is still ill-defined, because it comes out to f(2)=f(2). If you add f(2)=3 to the definition, then f(2) is well-defined (it's 3), but f(3) is not (we get f(3)=f(3), which is not a definition), and neither is f(4) (because it's equivalent to f(f(3)), which is not well-defined when f(3) isn't).

The phrase "smallest odd perfect number" corresponds to a well-defined number if and only if there is an odd perfect number; you do, however, need to define a perfect number previously. The phrase "smallest semiprime exceeding 10^2015" necessarily corresponds to a well-defined number, as it can be easily proven that there is at least one semiprime exceeding 10^2015; it doesn't matter whether anyone ever found the actual number. We can, similarly, talk about a number resulting from taking the decimal digits numbered 10^100 to 10^101 inclusive from the left of 4^4^4^4; it's clearly a well-defined number (regardless of the clear impossibility of it ever being calculated).

Consider now the phrase "largest number not exceeding 1000^1000^1000 without the letter 'a' in its name". This, as written, is ill-defined without specifically saying which system is used for the number names (for example, Conway-Wechsler, Knuth, or any variation thereof). It immediately becomes well-defined as soon as a particular unambiguous system is chosen, however (as it happens, the answer would be equal to 1000^1000^1000 under straight Conway-Wechsler, but a much smaller 2973-digit number under the Landon Curt Noll variation). The situation for the Bowers notation (beyond some particular point) is similar; there are sometimes several competing ways in which it could be expanded to result in an answer, and sometimes no way at all is evident from the rules. If the rules were unambiguous, the notation would indeed result (in each case) in some specific well-defined number. For what it's worth, Bowers is not alone here... my attempts at deciphering the notation used by Kharms in his classic 1931 googological treatise also ended similarly (as in, a structure was used that was not actually defined previously).