Potential vs Actual Infinity

Are there infinitely many numbers all at once, or can we only ever reach more and more of them without end? Joel David Hamkins traces this fundamental question from ancient geometry and Archimedes through Galileo's arguments for actual infinity, arriving at the modern landscape of potentialism and actualism. He examines ultrafinitism, the radical view that very large numbers may not meaningfully exist, and the famous challenge of drawing a line between numbers that exist and those that don't. The lecture then takes a precise turn, showing how modal logic can formalize different varieties of potentialism. Using axiom systems like S4, S4.2, and S4.3, Hamkins distinguishes linear, convergent, and radically branching forms of potentialism, culminating in a potentialist translation theorem. He closes by explaining how actualism, the view that infinite totalities are fully real, became the dominant framework in modern mathematics, with infinities built upon infinities as routine practice.