Löwenheim-Skolem and interpolation theorems in infinitary languages
David W. Kueker · Bulletin of the American Mathematical Society · 1972
Let L be a first-order finitary predicate language with equality.For each pair of infinite cardinals K and X with fc^/lwe let L KX be the logic extending L which allows the conjunction ( A ) and disjunction ( v ) of fewer than K formulas and the simultaneous universal or existential quantification of fewer than k variables.We set L^x -\J K L Kk .The standard syntactical and semantical concepts are defined as usual (see [1], [2]).If 0 is a sentence we write 211= 0 to mean that 0 is true on the model 21. 31 = Kk 23 means that 2Ï and 93 have the same true sentences of L KA .21,93, and 21, are always used for models for L, and we follow the convention that their universes are A, B 9 A t respectively.The cardinality of a set X is denoted by \X\.If L' is some other language, then L Kk is the corresponding infinitary logic built on L'.For ease in stating many of our results we assume, except in the last section, that L has only countably many nonlogical symbols.A detailed presentation of these and related results is in preparation for publication elsewhere.LEMMA.D is a countably complete filter, and if X^ e D for all Ç < K then {s:seXçfor all l;es}eD.