We might well believe that our physical universe is finite; but mathematics appears to posit infinite entities, such as the collection of the natural numbers. This raises a fundamental question in the philosophy of mathematics: To what extent and in which sense do mathematical infinities really exist? This question can be broken up in the following sub-questions: 1. Do potentially infinite collections exist? 2. Do actually infinite collections exist? 3. Do the actually infinite collections themselves form a potentially infinite collection of infinities of different sizes? 4. Is there a maximally large actual infinity? I will argue that it is rational to answer 'yes' to each of these questions.
