This paper discusses the detection of activations in statistical parametric maps (SPMs) from functional neuroimaging, particularly fMRI and PET. It introduces a hierarchy of tests based on the level of inference (voxel level, cluster level, and set level) and evaluates their relative sensitivity. The authors derive the probability of obtaining $c$ or more clusters with $k$ or more voxels above a threshold $u$ using distributional approximations from Gaussian field theory. The key contribution is the concept of set-level inference, which assesses the likelihood of observing a set of clusters that are unlikely to occur by chance. Set-level inferences are generally more powerful than cluster-level inferences, which are in turn more powerful than voxel-level inferences. The power of these tests varies with the spatial extent of the underlying signal and the signal-to-noise ratio. For distributed signals, set-level inferences are most powerful when the extent threshold matches the expected cluster size, while voxel-level inferences are most powerful when the signal is smaller than the resolution. The paper also discusses the interpretability of set-level inferences and their potential advantages in fMRI, where they can provide more comprehensive descriptions of activation patterns.This paper discusses the detection of activations in statistical parametric maps (SPMs) from functional neuroimaging, particularly fMRI and PET. It introduces a hierarchy of tests based on the level of inference (voxel level, cluster level, and set level) and evaluates their relative sensitivity. The authors derive the probability of obtaining $c$ or more clusters with $k$ or more voxels above a threshold $u$ using distributional approximations from Gaussian field theory. The key contribution is the concept of set-level inference, which assesses the likelihood of observing a set of clusters that are unlikely to occur by chance. Set-level inferences are generally more powerful than cluster-level inferences, which are in turn more powerful than voxel-level inferences. The power of these tests varies with the spatial extent of the underlying signal and the signal-to-noise ratio. For distributed signals, set-level inferences are most powerful when the extent threshold matches the expected cluster size, while voxel-level inferences are most powerful when the signal is smaller than the resolution. The paper also discusses the interpretability of set-level inferences and their potential advantages in fMRI, where they can provide more comprehensive descriptions of activation patterns.