In chapter 1 of Spivak's Calculus text he lays out some fundamental axioms of the integers. For instance that: $a \cdot 1 = a$, for all $a$. However he doesn't list an axiom that for instance says: $a \cdot 0 = 0$, for all $a$. This seems a bit arbitrary. Can we derive $a \cdot 0 = 0$ from Spivak's other axioms? On page $6$ he just says that $a \cdot 0 = 0$, for all $a$, without explanation.
Also he seems to be taking for granted that if $a = b$, then $a + c = b + c$. Another implicit axiom.
Why doesn't he mention these “implicit axioms” explicitly?