Putting my fields in order

I'm back to reading Analysis With an Introduction to Proof, 4th Ed. by Steven R. Lay. This is the book with the proof of the proof by the method of induction that I mentioned the other day. So I've made it through chapter 3, section 11: Ordered Fields relatively quickly largely because this is pretty much all review.

The interesting thing about ordered fields is that all of you are familiar from high school algebra with the things that determine whether something is a field and whether the field is ordered. It's just that nobody bothered to tell you this. An oversight I'm about to correct.

"What," you ask, "are these ordinary things that define fields?" Axioms. The very same axioms you learned in high school. And, yes, I'm going to tell you what they are, but first you need to learn a couple of mathy symbols. ℝ - The set of real numbers, and -the symbol meaning "is a member of" so x ℝ reads "x is a member of the set of real numbers. That's all you need to know.

On to the axioms.

There are 5 axioms for addition.

A1: For all x, y ℝ, x + y ℝ andif x = w and y = z, then x + y = w + z

A2: For all x, y ℝ, x + y = y + x

A3: For all x, y, z ℝ, x + (y + z) = (x + y) + z

A4: There is a unique real number 0 such that x + 0 = x for all x ℝ

A5: For each x ℝ there is a unique real number -x such that x + (-x) = 0.

There are also five axioms for multiplication that will look pretty much like the addition axioms.

M1: For all x, y ℝ, x ^. y ℝ andif x = w and y = z, then x ^. y = w ^. z

M2: For all x, y ℝ, x ^. y = y ^. x.

M3: For all x, y, z ℝ, x ^. (y ^. z) = (x ^. y) ^. z

M4: There is a unique real number 1 such that 1 ≠ 0 and x ^. 1 = x for all x ℝ.

M5: For each x ℝ with x ≠ 0, there is a unique real number 1/x such that x ^. 1/x = 1. You can write 1/x as x^-1.

Now for the last 5 axioms. You should recognize the first of these since it is the distributive law. This is pretty cool because it shows how addition and multiplication are related to each other.

D1: For all x, y, z ℝ, x ^. (y + z) = x ^. y + x ^. z

Then you have the 4 axioms that define order.

O1: For all x, y ℝ, exactly one of the relations x = y, x > y, or x < y holds (trichotomy law)

O2: For all x, y, z ℝ, if x < y and y < z, then x < z

O3: For all x, y, z ℝ, if x < y, then x + z < y + z

O4: For all x, y, z ℝ, if x < y and z > 0, then x . z < y . z

Actually, any set of numbers or mathematical system where the first 11 axioms are true is a field. If all 15 axioms are true, you have an ordered field. The set of rational numbers makes up an ordered field, but the set of integers does not (do you see why?).

Reference: Lay, S.R. (2005). Analysis With an Introduction to Proof, 4th Ed. Pearson-Prentice Hall: Upper Saddle River, NJ. pp 108 - 112.

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