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:

- Do potentially infinite collections exist?
- Do actually infinite collections exist?
- Do the actually infinite collections themselves form a potentially infinite collection of infinities of different sizes?
- Is there a maximally large actual infinity?

I will argue that it is rational to answer ‘yes’ to each of these questions.

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