zlacker

>>tromp+(OP)
I know the purpose of this article, but let me be "that person"---how would you do arithmetic? Or compare them? ;-) The proposed scheme is essentially a very compact version of constructive real number, which has a well-known caveat of not always being comparable to other numbers (because two real numbers can be made arbitrarily close to each other without being equal). It would be interesting to design a constructive real number format that can avoid those pedantries.

>>lifthr+qa7
The standard way of comparing these unfathomably huge numbers is by placing them in the Fast Growing Hierarchy. In fact, the article does this with both wCubed and Graham's number to show that the former is much larger.

>>tromp+OD7
I mean, as others pointed out, a number format typically needs much more than just a representation. If there is a bit-size limitation any operation has to round any excess back into the limit. A computational representation based on Turing machines or lambda calculus---again, which is not very different from a constructive real number in usage---do not provide an easy way to do that. That wouldn't have been an issue if there was no bit-size limitation.

Okay, let's ignore arithmetics and just allow comparison. As you've said, a common practice is to normalize it into some standard notation with a well-founded ordering. But there is no mechanical way to convert (or even bound) a computational representation to such notation---the general approach is therefore to compute a difference and check its sign. Not really good when it can continue even after the heat death of universe...

Frankly speaking, I rather expected to see some improvement over Level-Index number systems [1], but it turns out that this post is completely unrelated to number formats. Otherwise it is good, hence my mild frustration here :S

[1] https://www.mrob.com/pub/math/altnum.html#li_sli