Abstract
	    The definability of the Sigma_4-acceptable ideals

			     Eberhard Herrmann
			Humboldt University, Berlin

Harrington introduced the notion of Sigma_k-acceptable ideal. (These are special
ideals in the set of splitting halves of a computably enumerable set).
He showed that all 3-acceptable ideals are elementarily definable 
in the lattice of computably enumerable sets and proved the inductive step k
to k+2 for every k. Together, these imply that the Sigma_k-acceptable ideals 
for all odd k greater or equal to 3 are elementarily definable in the lattice.
We will show that also all 4-acceptable ideals are elementarily definable 
in this lattice, which together with the inductive step gives that all
Sigma_k-acceptable ideals for all even k greater or equal to 4 are elementarily 
definable, hence that for all k greater or equal to 3, the Sigma_k-acceptable 
ideals are definable.