The Artin‐Schreier theorem says that every formally real field has orders. Friedman, Simpson and Smith showed in [6] that the Artin‐Schreier theorem is equivalent to ${\mathsf{WKL}}_0$ over ${\mathsf{RCA}}_0$. We first prove that the generalization of the Artin‐Schreier theorem to noncommutative rings is equivalent to ${\mathsf{WKL}}_0$ over ${\mathsf{RCA}}_0$. In the theory of orderings on rings, following an idea of Serre, we often show the existence of orders on formally real rings by extending pre‐orders to orders, where Zorn's lemma is used. We then prove that “pre‐orders on rings not necessarily commutative extend to orders” is equivalent to ${\mathsf{WKL}}_0$.

中文翻译：

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可计算环上的订单

Artin-Schreier定理说，每个形式上的实场都有阶。Friedman，Simpson和Smith在[6]中证明了Artin-Schreier定理等于${\mathsf{WKL}}_0$ 过度 ${\mathsf{RCA}}_0$。我们首先证明Artin-Schreier定理对非交换环的推广等同于${\mathsf{WKL}}_0$ 过度 ${\mathsf{RCA}}_0$。在环的有序理论中，遵循Serre的思想，我们经常通过将预序扩展到使用Zorn引理的阶来证明形式实环上的阶的存在。然后，我们证明“环上的预购不一定能交换到订单”等同于${\mathsf{WKL}}_0$。