User blog comment:Alper2006/Proof that numbers are way bigger than you think/@comment-30754445-20180301091137

Beyond "unwritable" your categories seem to be completely arbitrary.

What do you mean by "unimaginable" and "undescribable"?

For example, a Decker (10^^10) is quite easily describable as a power tower of 10 tens. If the use of power towers here bothers you, then we can describe a Decker with 10 layers of zeros (as seen in the nice picture here).

The same goes for most of the named large numbers. It's just that the descriptions gradually become more and more complex. Using your limit of 100000 for the other categories, we should call any number than can be described in less than a million words "describable".

The threshold of the "indescribable" category would strongly depend on two things:

(1) The concepts we're permitted to use to describe the numbers without explaining them first.

(2) The amount of detail you require from the description. Do you require a complete description of the decimal expansion of the number (if so, then even 2^1000 is "undescribable")? Do you simply require a step-by-step method of generating the number (if so, then everything on your list except Rayo would be "describable").

And the same goes for the category of "unimaginable". What, exactly, are you requiring us to "imagine" about the number? If you're talking about the actual magnitude, then I argue that even a googol is completely unimaginable. If, on the other hand, you're talking about being able to place the number on some mental number line and having an intuitive way for comparing two such numbers, then - again - everything on your list (with the possible exception of Rayo) is "imaginable".

BTW you can translate the terms "imaginable" and "describable" to precise mathematical properties. The cut-off points would, of-course, depend on the mathematical system we're using to define the numbers (they're usually very very very large, though).