This paper studies filters on subsets of x which behave like ultrafilters on certain definable over L, subsets of an ordinal x 2 w. In particular it examines A,, IT, and 2, ultrafilters on subsets of x . The major result of section 2, theorem 2.4, is due to M. KAUFMAKK (151, theorem 1). However the proof presented here is different than his and uses the characterization of &+,-admissible ordinals given in theorem 1.4. In section 3 filters on subsets of x will be studies which are closed under certain "definable" diagonal intersections of subsets of x . Section 4 considers ultrapowers on L, modulo I7,, and Z,l ultrafilters of subsets of x . These ultrapowers consist of equivalence classes of partial Zri(Lx) functions f, whose domain belongs to the given IT, (or .Zn) ultrafilter. This leads to an "ultrapower" characterization of those x , for which there exists a &ultrafilter on subsets of x (theorems 4.6, 4.7).
I would like to thank W.RICHTER, K.PRIKRY, G.EUHRKEN and M. KAUFMANN for many helpful discussions on the contents of this paper.
1. Preliminaries and notation
Thih paper considers structures YJl = ( M , E ) and lil = ( N , li') for the language {E].
For such structures !Ill, %, ! I l l <,& lil is defined by M E N and YJt, lil satisfy the same , T, # sentences with parameters in YJl . % is said to be a n end extension of !Ill, and write
The symbol ! I l l <,,., ' % (i.e. % is a proper 2, end extension of %fl) is defined by YJt se and $2 <,, 8. Finally ! I l l s,# 111 means that the structure !Ill is isomorphic to a structure n' such that !Ill' <, 9. Z,t and IT,, formulas (n => 0) are defined as usual. A relation R g M A is Zrl('!JJ?) (respectively D,(YJl)) if there exists a En (respectively n,) formula ~( u , .
. . . , u,, zl, . . ., 2,) and elements a,, . . ., a, E M such that R = {(bl, . . ., b,) E E ilP: zln k ~( b , , . . ., b,, a,, . . ., a,)}. A,,(YJt) is defined by A,(%fl) = Z, (' i XX) AIT,('~XX). KP is the KRIPKE-PLATEK Set Theory (see [I]). (La: LY E Ord) denotes the hierarchy of constructible sets and L = u ( L a : LY E Ord}. For m a = a, < L e is the canonical well-ordering of L, (see [3], lemma 28). Also recall that for n 2 1, the satisfaction predicate .I=? (respectively kf;) is I7, (respectively &) over L, uniformly for all limit 01 2 co (see [3], lemma 9). An immediate consequence of JENSEN'S uniformization theorem is the folIowing (see [Ill).