In §5.3, we noted that type distinctions among various elementary objects were needed in order to avoid two linguistic problems. The first of these was notational collision; we saw that if we took, say, natural numbers and sets to be of the same type, then our theory incorrectly predicted that expressions like ‘1 + 1’ had multiple readings. Constructing this type distinction was hard because the standard foundational account defines natural numbers as sets. The second problem was that of unwarranted distinctions; we saw, for example, that the actual use of mathematical language required real numbers and complex numbers to be of the same type in order to assign a reading to an expression such as ‘
. < 1’. Preventing a type distinction here was difficult because the standard foundational account defines complex numbers as particular structures containing real numbers.