# Prove that the real number system is the only complete ordered field

In Rosenlicht's Intro to Analysis book, he states that

Since the addition and multiplication of decimals follow the usual rules of arithmetic, the real number system is completely determined by Properties I - VII, in the sense that if we have another triple $\{\mathbb{R}',+',\cdot'\}$ satisfying these properties then there will exists a unique one-one correspondence between $\mathbb{R}$ and $\mathbb{R}'$ preserving sums and products.

(The Properties I - VII mentioned are the axioms of a complete ordered field, 5 field axioms, one order axiom and one completeness axiom)

I don't see how you are supposed to make the jump from (addition and multiplication of decimals follow usual rules of arithmetic) $\implies$ ($\mathbb{R}$ is essentially the only complete ordered field). How would I go about proving this? Would I need to somehow find a correspondence between $\mathbb{R}$ and $\mathbb{R}'$ that preserves sums and products?

• It might help to know what Properties I - VII are. – Nick Peterson Nov 18 '16 at 21:19
• @NickPeterson First five properties are field axioms, sixth is order and seventh is completeness (least upper bound), so just the axioms for a complete ordered field – Joshua Lin Nov 18 '16 at 21:26
• Yes, you would need to find such a correspondence. This result is not trivial and its proof is, perhaps, a bit too technical to really be of interest in Rosenlicht's book (especially since he doesn't construct $\mathbb{R}$ starting from the axioms of the ZF theory of sets). IIRC, you can find a proof of what you want in this book. – Guest Nov 18 '16 at 22:02