Source: J. Symbolic Logic
Volume 74, Issue 2
Hirschfeldt and Shore have introduced a notion of stability for infinite
posets. We define an arguably more natural notion called weak stability, and
we study the existence of infinite computable or low chains or antichains,
and of infinite Π01 chains and antichains, in infinite computable
stable and weakly stable posets. For example, we extend a result of
Hirschfeldt and Shore to show that every infinite computable weakly stable
poset contains either an infinite low chain or an infinite computable
antichain. Our hardest result is that there is an infinite computable weakly
stable poset with no infinite Π01 chains or antichains. On the other
hand, it is easily seen that every infinite computable stable poset contains
an infinite computable chain or an infinite Π01 antichain. In Reverse
Mathematics, we show that SCAC, the principle that every infinite stable
poset contains an infinite chain or antichain, is equivalent over RCA0 to
WSCAC, the corresponding principle for weakly stable posets.
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