Real questions about infinity and probability

One may have the impression that the kinds of questions I like to ask about infinity and probability, questions involving zero probability events and infinitesimals are all purely hypothetical. We don’t actually play games with infinitely many die rolls and the like.

This is a mistake. Here are five non-hypothetical questions that have problematic features dealing with infinity:

1. How epistemically likely is it that we live in a multiverse with exactly K universes (where K is finite or infinite, and at least 1)?

2. How epistemically likely is it that we live in a multiverse that includes at least one K-dimensional universe (where K is finite or infinite, but different from 4)?

3. How epistemically likely is it that there are exactly K objects in existence (where K is finite or infinite, and at least around 10⁸⁰)?

4. How epistemically likely is it that in the future I will roll a die infinitely often and each time get heads?

5. How epistemically likely is it that the first counterexample to Goldbach’s conjecture is between 10¹⁰⁰ and 10¹⁰¹?

Since the epistemic probability that we live in an infinite multiverse is non-zero, questions 1 and 2 have non-hypothetical bite. Question 3 obviously has bite. Likewise, question 4 has bite because the epistemic probability of an infinite afterlife is non-zero. Question 5 has bite because it is epistemically possible that Goldbach’s conjecture is false.