The article investigates the mathematical and philosophical backdrop of the digital ocean as contemporary model, moving from the digitalized ocean of Georg Cantor’s set theory to that of Alan Turing’s computation theory. It examines in Cantor what is arguably the most rigorous historical attempt to think the structural essence of the continuum, in order to clarify what disappears from the computational paradigm once Turing begins to advocate for the structural irrelevance of this ancient ground.