[Someone]

> arithmetic is unary, and staying within the brainfuck virtual machine doesn't provide a way to do any better than unary arithmetic.

It won’t be efficient (but who cares about that in a true Turing machine?), but you can do binary/octal/decimal/whatever, using a series of cells each containing a number in the right range.

As an optimization, you can postpone any carries until you want to compare or print such numbers. Addition would ‘just’ loop over the cells in the numbers to be added, adding cells pairwise. For example, binary 11001 + 10101 would yield 21102.