next steps

Explanation of the foundation of mathematics has been completed.

We have gotten numbers and basic rules of calculation and logics on sets.

However, these are not enough, because every sets do not satisfied.
Especially rules of calculation depend on sets.

Numbers can exist in every sets, as whose elements are empty set. 

Real numbers satisfies all calculation rules.
Integers satisfies rules of addition, subtraction, and multiplication.

From next posts, we are going to look for the properties of sets
of which calculation rules are limited.