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