A NOTE ON REAL SUBSETS OF A RECURSIVELY SATURATED MODEL
by ATHANASSIOS TZOWARAS in Thessaloniki (Greece) 9 0. Introduction Let L = [+, ., I , <, 0} be the language of Peano Arithmetic (PA) and let M be a countable recursively saturated model of PA. For every a E M let -a be the equivalence x -a y ++ for all p(vo, v,) E L , M k pfx, a ) * p(y, a ) . The sets X S M which consist of whole equivalence classes with respect to some -a are called real. The nonreal sets are called imaginary.
There is an old and well-known criterion for real sets due to D. KUEKER and G. E. REYES (see [4], Theorem 1.5) which roughly says that real sets are those with few automorphic images.
Theorem 0.1. (KUEKER-REYES). Let M be countable. r f M has uncounrably many automorphisms, then X E M is red iff the set (f"X: fe Aut ( M ) } is countable.
As far as we know there had been no further interest on the properties of the sets with few and many autornorphic images, until this division reappeared, possibly independently, some years later, in the work of members of the Prague School concerning the Alternative Set Theory. In particular the terms "real" and "imaginary" were used by K. CUDA and P. VOPENKA in 197-202.