Axioms for the reals

Kent Merryfield

It is common to develop the real numbers axiomatically. That is, we take $\mathbb{R}$ to be a set which satisfies the axioms of an ordered field, and which also satifies one additional axiom. What should that one additional axiom be? I offer seven possible choices:

1. Every nonempty subset that is bounded above has a least upper bound.

2. Suppose $A\cup B=\mathbb{R}, A\cap B=\emptyset, A\ne\emptyset, B\ne\emptyset$ , and for all $a\in A,b\in B, a<b$ , then either $A$ has a greatest element or $B$ has a least element.

3. Monotone bounded sequences converge.

4. Cauchy sequences converge.

5. If $I_1\supset I_2\supset I_3\supset\cdots$ are nested closed bounded intervals, then $\bigcap_{n=1}^{\infty}I_n\ne\emptyset$ .

6. Bounded sequences have convergent subsequences.

7. Every decimal expansion converges to a real number, and every real number has at least one decimal expansion.

(Note: the second phrase in #7 doesn't look like completeness, but without it, we wouldn't have the Archimedean property.)

The most common textbook path through this would start by assuming 1 as an axiom. I would consider the following implications "textbook obvious":
$1\implies 3\implies 5\implies 6\implies 4, 1\implies 2, 3\implies 7.$

I have two challenges:

First, to show that all seven of these statements are equivalent; that we could start with any one as the axiom and prove the other six as theorems.

Second, to prove as many of the 42 direct implications as possible without going through one of the other statements. Clearly, you can't do all of these, but which ones can you do?

Any construction of the real numbers should lead fairly quickly to one of these properties.

**Posted:** Sat Oct 09, 2004 8:38 am

dickchimney · Poincare Conjecture Offline Joined: 06 Mar 2004 Posts: 218

What about the Dedekind method ( Dedekind slice )of constructing real numbers??

( A real number is a set of rational numbers with some properties )

**Posted:** Sun Jan 02, 2005 1:11 am

ZetaX

The dedekind methode is very similar to/exactly the same as the second axiom.
(Dedekind axioms: a dedekindian slice is a (ordered) pair $(A,B)$ of two sets of rational numbers that fulfil $A\cup B=\mathbb{Q}, A\cap B=\emptyset, A\ne\emptyset, B\ne\emptyset, \forall a \in A, b \in B : a<b$ and $B$ has no least element).
But I think there is an other problem: not all these axioms guarant the Archimedean property (for example the 4.)

**Posted:** Sun Jan 02, 2005 2:17 am