I've been trying to study the field axioms in order to eventually go through Spivak's Calculus (I'm not a math major, but just interested). I noticed that different books have different axioms.
For example, Spivak lists the order properties as:
- $a = 0$, $a$ is in $P$ (positive numbers), or $-a$ is in $P$
- $a$ and $b$ are in $P$ $\implies$ $a+b$ is in $P$
- $a$ and $b$ are in $P$ $\implies$ $ab$ is in $P$
He then defines $a>b$ as meaning $a-b$ is in $P$.
However, another book I looked over stated the order axioms as:
- either $a = b$, $a<b$, or $b<a$
- $a<b$ and $b<c$ $\implies$ $a<c$
- $a<b$ $\implies$ $a+c < b+c$
- $a<b$ and $o<c$ $\implies$ $ac < bc$
Why is there a difference, and is one set of order axioms more basic or fundamental than the other?
Also, Spivak doesn't mention an axiom of closure for either addition or multiplication as one of the axioms, whereas the other book does. Can closure be proven based on the other axioms and is unnecessary?
I was just wondering which axioms are the best to take for granted. Thanks for any help!
Many things in math can be defined in multiple ways. Usually the different definitions are equivalent. You choose one definition, then prove the other properties as theorems. Once you have done that, it doesn't matter too much which definition you chose.