# Ordered field with nested intervals axiom without Archimedean axiom

Does an ordered field $F$ satisfying the Cauchy-Cantor axiom but not the Archimedean axiom exist?

Cauchy-Cantor axiom: Every system of nested intervals $I_1 \supset I_2 \supset \cdots \supset I_n \supset \cdots$ have a common point. Here, an interval $I$ is any set of the form $I= \{ x \in F \mid a \leqslant x \leqslant b\}$ for some $a\le b$ in $F$.

Archimedean axiom: For every $x,y \in F$ with $y>0$ there exists a unique $n \in \mathbb{Z}$ such that $ny \leqslant x < (n+1)y$.

• Start post has been changed. – kp9r4d Jul 17 '14 at 16:31
• Thanks for your answer! "does the nested interval property imply that an ordered field is Archimedean", - how can I prove that? – kp9r4d Jul 17 '14 at 17:10
• One can construct an example using formal Laurent series. For a discussion of every conceivable related question, please see this article by James Propp. – André Nicolas Jul 17 '14 at 17:16
• "The nested interval property statement has the inclusions running the wrong way.", - my fault, I changed. Thanks for article! – kp9r4d Jul 17 '14 at 17:21
• The question seems clear enough to me. – Michael Hardy Jul 17 '14 at 19:15