A COMPLETENESS RESULT FOR FIXED‐POINT ALGEBRAS

FPAs were introduced by C. SMORY~SKI in [lo]; these structures arise from the following consideration: let T be an axiomatic theory in which every recursive function is representable; let for every two T-sentences p , q, p w T q stand for t , p t, q, and let us call a formula Fx in one free variable extensional iff, for every two T sentences p and q, p q implies F@ , F c ~ ; then every extensional formula Fx induces a mapping P from the Lindenbaum sentence algebra 8, of T into a, defined by P [ p ] T ’= [F$]-T (here denotes the equivalence class of p modulo -,). Now, let Br = { P : Fx is extensional} (of course we identify P and d iff they coincide as mappings); if we define the Boolean operations in BT in an obvious way’) the pair (a,., 23,) becomes an FPA, called the full Lindenbaum FPA of T (clearly, Condition 4 is the algebraic translation of GODEL’S Diagonalization Lemma).