We study systemic risk in an interbank market, employing an explicit axiomatization inspired by Eisenberg and Noe (Manag Sci 47(2):236–249, 2001) and Rogers and Veraart (Manag Sci 59(4):882–898, 2013). Instead of focusing on a clearing payment scheme, we characterize the smallest (in the sense of inclusion) set of ex post defaulting firms. This novel approach allows us to analyze the normative implications of the Eisenberg–Noe axioms. We first show that the Absolute Priority axiom, which states that defaulting firms must end up with zero net worth, has no impact on minimal default sets. Second, relaxing the Limited Payments axiom, which can be interpreted as allowing a central planner to transfer resources from rich firms to poor, does not further reduce the minimal default sets, although other default sets are possible. Our normative analysis sheds new light on the possible impacts of clearing mechanisms on default outcomes.