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29 March 2024
 
  » arxiv » math.LO/0407405

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Half of an inseparable pair
Arnold W. Miller ;
Date 23 Jul 2004
Subject Logic MSC-class: 03E15, 03E35, 03E60 | math.LO
AbstractA 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/
Source arXiv, math.LO/0407405
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