Basically, mathematics is founded by some axioms of sets.
(1)empty set
(2)extensional
(3)pairing
(4)union of sets
(5)power sets
(6)separation
(7)replacement
(8)regularity
(9)infinite
(10)axiom of choice
I do not think every axioms are clear and definite.
Strictly, for example, by axiom of infinite and separation we will get axiom of empty set.
We are able to reduce ten axioms.
Next we got the various kinds of numbers,
natural numbers$\mathbb{N}$, integers$\mathbb{Z}$, rational numbers$\mathbb{Q}$, real numbers$\mathbb{R}$, complex numbers$\mathbb{C}$ on sets.
Then, we saw the basic rules of calculation and logics.
Modern mathematics is said to be enough by them.
