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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:

  1. $a = 0$, $a$ is in $P$ (positive numbers), or $-a$ is in $P$
  2. $a$ and $b$ are in $P$ $\implies$ $a+b$ is in $P$
  3. $a$ 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:

  1. either $a = b$, $a<b$, or $b<a$
  2. $a<b$ and $b<c$ $\implies$ $a<c$
  3. $a<b$ $\implies$ $a+c < b+c$
  4. $a<b$ 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!

celtschk
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Honus27
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  • Seems something went wrong with your editing: "either a = b, a a a+c < b+c" surely isn't what you wanted to write. – celtschk Mar 04 '17 at 16:37
  • I noticed that the content was recognizable in the source. I've edited it to be visible (and readable). Please next time make sure your posts can be read. – celtschk Mar 04 '17 at 16:48
  • Those are equivalent ways to define an ordered field, via an order relation or positive cone. This is explained in many places, e.g. see Wikipedia. As for closure under addition and multiplication, probably they are hypothesized to be binary operations on the set, which implies closure. Is all clear now? – Bill Dubuque Mar 04 '17 at 16:54

1 Answers1

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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.

Ross Millikan
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