I'm not familiar with what basal numbers means.

Anyway, 5555 is just one number in the decimal system that fulfills the requirement that the position of digits is irrelevant, whereas most decimal numbers do not. In the tally mark system all numbers fulfill this requirement.

However, the thing I like most about it is that you'll never need to prove that I+I=II. It literally is II.

I remember reading the book called "Gödel, Escher, Bach" which is about Gödels incompleteness theorem. At some point it comes across this kind of thing and demonstrates how any natural number is the successor of the previous number, basically defining numbers as tally marks. From there it goes on to demonstrate why math itself is incomplete. It's kinda a fat book, but if you're into numbers, logic and coding it's a must read.