I have just begun reading "Mathematical Analysis", $2^{nd}$ edition by Apostol. In the beginning of chapter $1$ we are introduced to $9$ axioms ($+1$ later). They are the field axioms and the order axioms (the later one which I haven't gotten to is the "completeness axiom" or "axiom of continuity"). However, on this website I found more, such as "There exists a unique number $0$ such that $a + 0 = a$ for any real number $a$."
Do the axioms outside of field axioms, order axioms, axiom of continuity have a special name? Why where they not even mentioned in the book?
This is the axiom of identity for addition, which states that there is an identity element (adding it does not change the value) for addition.
See if you can figure out the identity element for multiplication.