CANTOR SETS AND FIELDS OF REALS

Our main result is a construction of four families C1, C2, B1, B2 which are equipollent with the power set of R and satisfy the following properties. (i) The members of the families are proper subfields K of R where R is algebraic over K. (ii) Each field in C-1 boolean OR C-2 contains a Cantor set. (iii) Each field in B-1 boolean OR B-2 is a Bernstein set. (iv) All fields in C-1 boolean OR B-1 are isomorphic. (v) If K, L are fields in C2. B2 then K is isomorphic to some subfield of L only in the trivial case K = L.