# What are these axioms called?

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?

• Those would be called "basic properties" I think. You should be able to derive existence of additive identity $0$ from the axioms in your book. – AbstractAlgebraLearner Dec 18 '15 at 2:17
• – Cameron Buie Dec 18 '15 at 2:34

• Well the multiplicative identity would be x$1$, but I am wondering if these are really axioms or if they can be derived from the 10 axioms in my book, because I don't understand why the author wouldn't bother to include the identity of addition since it is so crucial – Ovi Dec 18 '15 at 2:20
• @Ovi There is no such thing as something being "really" an axiom. A set of axioms is just a starting point. Different authors may use different sets of axioms, and one author's axiom might be another author's theorem. Apparently the author of your textbook chose not to include $x+0=x$ as an axiom, but it should be derivable from the axioms that are included. – Ted Dec 18 '15 at 3:41
• @Ted I'd love to be able to start working on some proofs, but I don't really know what I'm allowed to work with. For example, axiom $4$ says that "there is always a z such that $x + z = y$, and x - x is denoted by $0$ (it can be proven that $0$ is independent of $x$.)" So I'm not even sure if at this point I'm allowed to work with zero as a real number, or if it's just a symbol/placeholder. And if it is a number, I don't see why you need a proof, that $0$ is independent of $x$, if by definition $x - x$ = $0$ and $x$ could be anything – Ovi Dec 18 '15 at 9:21