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Linearizability is the strongest known consistency property of shared objects. In asynchronous message passing systems, Linearizability can be achieved with ◊S and a majority of correct processes. In this paper we introduce the notion of Eventual Linearizability, the strongest known consistency property that can be attained with ◊S and any number of crashes. We show that linearizable shared object implementations can be augmented to support weak operations, which need to be linearized only eventually. Unlike strong operations that require to be always linearized, weak operations terminate in worst case runs. However, there is a tradeoff between ensuring termination of weak and strong operations when processes have only access to ◊S. If weak operations terminate in the worst case, then we show that strong operations terminate only in the absence of concurrent weak operations. Finally, we show that an implementation based on P exists that guarantees termination of all operations.