Is it possible to construct a real number theory on Peano arithmetic?

Question

I know how to construct $\mathbb{Z}, \mathbb{Q}, \mathbb{R}$ from $\mathbb{N}$ in set theory. For example, the construction of $\mathbb{Z}$ is, $$\mathbb{Z}=\mathbb{N}^2/\sim$$ $$(a, b)\sim(c, d)\Leftrightarrow a+d=b+c$$

However, I do not know how to construct tuples and quotients in PA.

Is it possible to construct $\mathbb{R}$ in Peano arithmetic?

Answer 1

See Peano Axioms, it's possible construct a real number theory