axiomatic sets 29 (restart)

For a long time, we discussed the number system or the structure of numbers.
It means that all numbers is sets.

As a function or a relation has already been defined by sets,
all statements in mathematics can be written by sets.

We shall return to the axiom of sets.

We have already addressed 6 axioms of sets.

1.empty set
There is a empty set.

2.extensionality
All elements is same if and only if the two sets are same.

3.pairing
There is a set such that if a element is in the set, the element is equal to one of two elements of the set.

4.union of sets
There is a set such that if a element is in the set, the element is in one of two other sets or in other both two sets.  

5.power sets
There is a set whose elements are all subsets of the set.
(subsets are made by the axiom of extensionality. )

6.separation
We are able to make a set whose elements are selected from a given set.
(for a selection, we need a formula or a statement of properties. )


There still remain several axioms in ZFC.

In next posts, we will discuss the axiom of replacement which is chosen behalf of the axiom of separation.