Is a “fully automated” capitalism possible? According to Marx's labor theory of value (LTV) and the theory of surplus-value derived from it, a fully automated economy cannot be profitable. To refute the theory, critics have put forth various thought experiments claiming to show that a fully automated but profitable capitalist economy is conceivable. I argue that the thought experiments fail to demonstrate conceivability, because they misunderstand the role of labor in a commodity economy. In the latter, labor is a means for acquiring the property of others, and so it cannot be eliminated so long as the economy is based on the commodity form. Quite apart from issues of technical feasibility, then, the idea of a fully automated commodity economy is conceptually incoherent. The ineliminability of labor under the commodity form reflects the limits to the socialization of production this form imposes.