In Intermediate Real Analysis by Emanuel Fischer page 6, the author states an axiom that says
(Axiom Q) If $x$ and $y$ are real numbers, where $z+y\neq z$ holds for some real $z$, then there exists a real number $q$ such that $yq=x$.
Is there a name for this axiom? I believe "$z+y\neq z$" is a typo; shouldn't it be $z+y\neq x$? Does anyone know if there is a list of typo for this book?
He's postulating the axiom that quotients exist for real numbers, with the necessary restriction that the dividend must be nonzero, i.e. it cannot be the addtive neutral element (usually denoted by $0)$.
Unlike the most common axiomatization of a field, the author does not include axioms for the neutral elements $0$ and $1$ for addition and multiplication. Rather, he defines the additive and multiplicative groups using axioms that differences and quotients exist, i.e. $\,a + x = b\,$ is solvable for all $\,a,b\in \Bbb R,\,$ and $\, a\cdot x = b\,$ is solvable for all $\,a,b\,$ when $\,a\,$ is not additively neutral (i.e. when $\, x + a = x\,$ fails to be true for all $\,x\in \Bbb R,\,$ i.e. when $\,r+a\neq r\,$ for some $\,r\in \Bbb R).$
Later he proves that there exist unique additive and multiplicative neutral elements, denoted by the customary symbols $0$ and $1$.
This axiomatization of groups using division (or subtraction) is less common than that using inverses and neutral elements, but it does occur from time to time (and if memory serves correct it has been discussed here in the past on multiple occasions, e.g. here). In fact there are many known axiomatizations of groups, rings, fields. Typically we choose one that proves convenient for the context at hand.
