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COUNTABLE ORDINALS AND THE ANALYTICAL HIERARCHY, I

This paper is a contribution to the study of projective sets under the hypothesis of definable determinacy. For our purposes here this can be understood as the hypothesis every set of rea1s in L[ω^ω] (= the smallest inner model of ZF containing ω^ω) is determined. Since questions about the projective hierarchy are absolute between the real world and 
L[ω^ω] it is innocuous to work entirely within L[ω^ω].