In modern operating systems and programming languages adapted to multicore computer architectures, parallelism is abstracted by the notion of execution threads. Multi-threaded systems have two major specificities: 1) new threads can be created dynamically at runtime, so there is no bound on the number of threads participating in a long-running execution. 2) threads have access to a memory allocation mechanism that cannot allocate infinite arrays. This makes it challenging to adapt some algorithms to multi-threaded systems, especially those that assign one shared register per process. This paper explores the synchronization power of shared objects in multi-threaded systems by extending the famous wait-free hierarchy to take these constraints into consideration. It proposes to subdivide the set of objects with an infinite consensus number into five new degrees, depending on their ability to synchronize a bounded, finite or infinite number of processes, with or without the need to allocate an infinite array. It then exhibits one object illustrating each proposed degree. CCS CONCEPTS • Theory of computation → Distributed computing models; • Software and its engineering → Process synchronization; • Computer systems organization → Multicore architectures; Dependable and fault-tolerant systems and networks.