## Definition ## ### Totally ordered abelian groups ### An abelian group $R$ is a totally ordered abelian group if it comes with a function $\max:R \times R \to R$ such that * for all elements $a:R$, $\max(a, a) = a$ * for all elements $a:R$ and $b:R$, $\max(a, b) = \max(b, a)$ * for all elements $a:R$, $b:R$, and $c:R$, $\max(a, \max(b, c)) = \max(\max(a, b), c)$ * for all elements $a:R$ and $b:R$, $\max(a, b) = b$ implies that for all elements $c:R$, $\max(a + c, b + c) = b + c$ * for all elements $a:R$ and $b:R$, $\max(a, b) = a$ or $\max(a, b) = b$ ### Strictly ordered pointed abelian groups ### A totally ordered commutative ring $R$ is a strictly ordered pointed abelian group if it comes with an element $1:R$ and a type family $\lt$ such that * for all elements $a:R$ and $b:R$, $a \lt b$ is a proposition * for all elements $a:R$, $a \lt a$ is false * for all elements $a:R$, $b:R$, and $c:R$, if $a \lt c$, then $a \lt b$ or $b \lt c$ * for all elements $a:R$ and $b:R$, if $a \lt b$ is false and $b \lt a$ is false, then $a = b$ * for all elements $a:R$ and $b:R$, if $a \lt b$, then $b \lt a$ is false. * $0 \lt 1$ * for all elements $a:R$ and $b:R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a + b$ ### Strictly ordered pointed $\mathbb{Q}$-vector space ### ... ### Archimedean ordered pointed $\mathbb{Q}$-vector space ### A strictly ordered pointed $\mathbb{Q}$-vector space $A$ is an Archimedean ordered pointed $\mathbb{Q}$-vector space if for all elements $a:A$ and $b:A$, if $a \lt b$, then there merely exists a rational number $q:\mathbb{Q}$ such that $a \lt h(q)$ and $h(q) \lt b$. ### Sequentially Cauchy complete Archimedean ordered pointed $\mathbb{Q}$-vector space ### Let $A$ be an Archimedean ordered pointed $\mathbb{Q}$-vector space and let $$A_{+} \coloneqq \sum_{a:A} 0 \lt a$$ be the positive elements in $A$. $A$ is **sequentially Cauchy complete** if every Cauchy sequence in $A$ converges: $$isCauchy(x) \coloneqq \forall \epsilon \in A_{+}. \exists N \in I. \forall i \in I. \forall j \in I. (i \geq N) \wedge (j \geq N) \wedge (\vert x_i - x_j \vert \lt \epsilon)$$ $$isLimit(x, l) \coloneqq \forall \epsilon \in A_{+}. \exists N \in I. \forall i \in I. (i \geq N) \to (\vert x_i - l \vert \lt \epsilon)$$ $$\forall x: \mathbb{N} \to A. isCauchy(x) \wedge \exists l \in A. isLimit(x, l)$$ category: not redirected to nlab yet