This paper is of heuristic and explorative character. Our aim is to explain a new conceptual tool for the study of infinitary logics (infinitary languages) and to illustrate its uses. For the former purpose, explicit formal definitions may be more of an obstacle than a help, and for the latter purpose strict proofs are not necessary, either, especially as our illustrations of the power of the new approach pertain to the systematization of known results rather than to the establishment of new ones. For these reasons, our discussion will be in this paper rather intuitive and discursive and will mostly dispense with strict definitions and proofs.
Our starting point is the observation that most currantly studied infinitary languages fetal.9, certain finitistic features which are not needed either for syntactical or semantical purposes. RougiAy speaking, infinilary formulas are still thought of as having been built up by finite or transfinite recursion from atomic formulas (see e.g. [ 1 ] ). This tends to impose various finitistic features on tile resulting formulas. For instance, if the only infinitary element in a language is a rule for forming infinite conjunctions and disjunctions, the foP(~wing finiteness properU, holds in it: (F. 1 ) Any descending chain of proper subformutas is finite.