User blog comment:P進大好きbot/What does a computable large number mean?/@comment-4224897-20180610135217

The way I see it, the concept of "computable numbers" is meaningless for the reasons you said, but computable googolisms are a concept that actually makes sense. The reason for this is because the word "googolism" referes to any name given to a number, and those names are usually accompanied by definitions for numbers.

For example, consider the busy beaver function, a function known to be uncomputable, and the number 13, the value the function outputs when 4 is input. The googolism "thirteen" for this number is computable, since knowing the English language it would be defined as "10 + 3". However, if we give a googolism defined as BB(4) (let's call it tetrabeaver, even though that name sucks), it would be uncomputable; even though the value of "tetrabeaver" is known to be 13, that value cannot be derived using a computer program.

It should be obvous that there isn't really a hard bound to what large numbers could and couldn't be derived using computable functions. Rayo's number is defined by plugging a googol into Rayo's function because a googol is much larger than the amount of symbols that can be stored in the observable universe, so that number will probably be larger than any number humans could devise using computable functions. BIG FOOT takes that to an even further extreme, by iterating the amount of symbols used upon itself ten times to make it extra safe it's larger than computable googolisms humans can devise.

I hope this helps! I've been out of touch with googology lately, so hopefully my explanation makes sense.