Our principal result is that there exist two incomparable recursively enumerable degrees whose greatest lower bound in the upper semilattice of degrees is 0. This was conjectured by Sacks . As a secondary result, we prove that on the other hand there exists a recursively enumerable degree a < 0(1) such that for no non-zero recursively enumerable degree b is 0 the greatest lower bound of a and b.
The proof of the main theorem involves a method that we have developed elsewhere  to deal with situations in which a partial recursive functional may interfere infinitely often with an opposed requirement of lower priority.
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