Definition of rational numbers from real numbers

Question

Usually the set of numbers are introduced starting from integers, from wich the rational numbers are defined using equivalence classes of couples of integer numbers. Than, from these rational numbers, we can construct the real numbers via Dedekind cuts or Cauchy sequences.

But real numbers can be defined in purely abstract way as a set that is a field with a total order that is Dedekind- complete (we can refer to Tarski's axiomatization).

Is it possible, starting from this abstract definition of reals, define the subfield of rationals (and the subset of integers)?

In other words can we define the usual sets of numbers starting from the real numbers instead of from integers?

Answer 1

Yes, of course. Because $\mathbb R$ is a field, it has a unit, $1$. Then you can immediately construct the integers by considering the additive subgroup generated by $1$: you have $0$, $1+2$, $2+1=3$, etc., and $-1$, $-1+(-1)=-2$, etc.

And now you can consider the subfield of $\mathbb R$ generated by $\mathbb Z$: that requires you to have the multiplicative inverses of the nonzero integers: $1/n$, for all $n\in\mathbb Z$, and $m/n=1/n+\cdots+1/n$ for all $m\in\mathbb N$.

Answer 2

We have an embedding $\Bbb Z \hookrightarrow \Bbb R$.

Define the set of rational numbers as

$\tag 1\Bbb Q = \{ab^{-1} \, | \, a,b \in \Bbb Z \text{ with } b \ne 0 \}$.

The set $\Bbb Q$ is a subfield of $\Bbb R$.

It is closed under addition:

$\quad ab^{-1} + cd^{-1} = (ad + cb) {(bd)}^{-1}$

etc.

Answer 3

$\mathbb Q$ is the smallest subfield of $\mathbb R$.