The Cauchy real numbers $\mathbb{R}_C$ is a set with a function $\iota \in {\mathbb{R}_C}^{C(\mathbb{Q})}$, where $C(\mathbb{Q})$ is the set of Cauchy approximations in $\mathbb{Q}$ and ${\mathbb{R}_C}^{C(\mathbb{Q})}$ is the set of functions with domain $C(\mathbb{Q})$ and codomain $\mathbb{R}_C$, such that

where the premetric for $\epsilon:\mathbb{Q}_{+}$ is defined as

In homotopy type theory

Let $R$ be a dense integral subdomain of the rational numbers $\mathbb{Q}$ and let $R_{+}$ be the positive terms of $R$.

The Cauchy real numbers $\mathbb{R}_C$ are inductively generated by the following:

• a function $\iota: C(R, R_{+}) \to \mathbb{R}_C$, where $C(R, R_{+})$ is the type of $R_{+}$-indexed nets in $R$ that are $R_{+}$-Cauchy approximations.

• a dependent family of terms

  $$a:C(R, R_{+}), b:C(R, R_{+}) \vdash eq_{\mathbb{R}_C}(a, b): \left( \prod_{\epsilon:R_{+}} a_\epsilon \sim_{\epsilon} b_\epsilon \right) \to (\iota(a) =_{\mathbb{R}_C} \iota(b))$$

  where the $R_{+}$-premetric for $\epsilon:R_{+}$ is defined as

  $$a \sim_{\epsilon} b \coloneqq \vert a_\epsilon - b_\epsilon \vert \lt 4 \epsilon$$

• a family of dependent terms

  $$a:\mathbb{R}_C, b:\mathbb{R}_C \vdash \tau(a, b): isProp(a =_{\mathbb{R}_C} b)$$

Comment

The Cauchy real numbers are sometimes defined using sequences $x:\mathbb{N} \to R$, $y:\mathbb{N} \to R$ with modulus of Cauchy convergence $M:R_{+} \to \mathbb{N}$, $N:R_{+} \to \mathbb{N}$ respectively. However, the composition of a sequence and a modulus of Cauchy convergence yields a Cauchy approximation, so one could define Cauchy approximations $a:R_{+} \to R$ as $a \coloneqq x \circ M$ and $b:R_{+} \to R$ as $b \coloneqq y \circ N$, with $x_{M(\epsilon)} = a_\epsilon$ and $y_{N(\epsilon)} = b_\epsilon$ following from function evaluation. In fact, Cauchy approximations and sequences with a modulus of Cauchy convergence are inter-definable.