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?

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    $\begingroup$ It might help to know what Properties I - VII are. $\endgroup$ – Nick Peterson Nov 18 '16 at 21:19
  • $\begingroup$ @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 $\endgroup$ – Joshua Lin Nov 18 '16 at 21:26
  • $\begingroup$ 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. $\endgroup$ – Guest Nov 18 '16 at 22:02

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