Half of an inseparable pair

Details Half of an inseparable pair Half of an inseparable pair

2004-07-23

arxiv.org/abs/math/0407405v1

Mathematics

Logic

LaTex2e 16 pages

Scientific paper

A classical theorem of Luzin is that the separation principle holds for the Pi^0_alpha sets but fails for the Sigma^0_alpha sets. We show that for every Sigma^0_alpha set A which is not Pi^0_alpha there exists a Sigma^0_alpha set B which is disjoint from A but cannot be separated from A by a Delta^0_alpha set C. Assuming Pi^1_1-determancy it follows from a theorem of Steel that a similar result holds for Pi^1_1 sets. On the other hand assuming V=L there is a proper Pi^1_1 set which is not half of a Borel inseparable pair. These results answer questions raised by F.Dashiell. Latest version at: www.math.wisc.edu/~miller/

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