CANTOR SETS AND FIELDS OF REALS

Abstract

Our main result is a constructionof four families ${\mathcal{C}}_1,{\mathcal{C}}_2,{\mathcal{B}}_1,{\mathcal{B}}_2$which are equipollent with the power set of $\mathbb{R}$and satisfy the following properties.(i) The members of the families are proper subfields $K$ of $\mathbb{R}$where $\mathbb{R}$ is algebraic over $K$.(ii) Each field in ${\mathcal{C}}_1\cup {\mathcal{C}}_2$ contains a {\it Cantor set}.(iii) Each field in ${\mathcal{B}}_1\cup {\mathcal{B}}_2$ is a {\it Bernstein set}.(iv) All fields in ${\mathcal{C}}_1\cup {\mathcal{B}}_1$ are isomorphic.(v) If $K,L$ are fields in${\mathcal{C}}_2\cup {\mathcal{B}}_2$ then $K$ is isomorphic to somesubfield of $L$ only in the trivial case $K=L$.