Omitting Types: Application to Descriptive Set Theory

The omitting types theorem of infinitary logic is used to prove that every small II set of analysis or any small 2. set of set theory is constructible. In what follows we could use either the omitting types theorem for infinitary logic or the same theorem for what Grilliot[2] calls (eA)-logic. I find the latter more appealing. Suppose i_ is a finitary logical language containing the symbols of set theory as well as a constant symbol a tot each a in the transitive set A. For this language we will use only (cA)-models, that is to say, end extensions of the model (A, e). Corresponding to this restricted notion of model is a strengthened notion of proof, (eA)-logic. In addition to the usual finitary rules of proof, this logic contains rules Ra for each a in A. Rule R says "From tp(b) tot each b in a, you may conclude Vx £ aj < tox. Here a>? is the first ordinal not recursive in a. It has also been observed that S = Q, where Q is the set of a which are constructible by stage tu. in the constructible hierarchy. Since Q C L, in order to prove that no small II, set has a nonconstructible element, Received by the editors June 15, 1973. AMS (MOS) subject classifications (1970). Primary 02K30.