Abstract
As we have seen earlier, many basic structural properties of the projective sets of the first two levels are consequences of the fact that the classes Π 11 , Σ 11 have the scale property. Using Projective Determinacy, we will establish in this section that this property propagates throughout the projective hierarchy with a periodicity of order 2, so that we have the following picture:
where the boxed classes are scaled (and thus also satisfy the uniformization, rank, and generalized reduction properties) and the other classes satisfy the generalized separation property. Thus the basic structure of the projective hierarchy is periodic of order 2. However, a finer analysis reveals significant structural differences, for example, between the first and the higher odd levels (see A. S. Kechris, D. A. Martin, and R. M. Solovay [1983]), that we will not pursue here.
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© 1995 Springer-Verlag New York, Inc.
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Kechris, A.S. (1995). The Periodicity Theorems. In: Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol 156. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4190-4_39
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DOI: https://doi.org/10.1007/978-1-4612-4190-4_39
Publisher Name: Springer, New York, NY
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