Mathematics is not based on just the decimal system. The binary system can do anything the decimal system also does, but their foundations are not the same.

There is no 'the 1' in the binary system. Rather, there are many 1s.

The word Unity, for instance, is not a common idea in the binary system. One has to create the word, for instance, with 11100111, agreeing on that, and then from that moment on using it to express Unity. Meanwhile, in the decimal system, 1 can indeed represent Unity (the other case is 'unit').

From this, one can see that the decimal system comprises of accepted shortcuts. Instead of working with the basics of mathematics, the decimal system is an agreement on what certain numbers declare to be the truth. And as such, there is a fallacy included in the decimal system:

Only when we use the decimal system do we have mathematical outcomes that make sense when indeed declared as belonging within that system.

Mathematics as a whole is greater than the decimal system. Mathematics is any mathematical system, and no answer is correct unless the user declares which mathematical system was used.

Gödel was right. No axioms can be placed at the top level and present a completeness. All axioms belong to their own system.