Cantor sets and fields of reals
Abstract
Our main result is a construction of four families C1,C2,B1,B2 which are equipollent with the power set of the real line R and satisfy the following properties. (i) The members of the families are proper subfields of R whose algebraic closures equal the field C. (ii) Each field in C1vC2 contains a Cantor set. (iii) Each field in B1vB2 is a Bernstein set. (iv) All fields in C1vB1 are isomorphic. (v) If K,L are fields in C2vB2 then K is isomorphic to a subfield of L only in the trivial case K=L.
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