You know for sure that infinitely many people, including yourself, each are independently tossing fair coins. You don't see your coin's result. But then you learn for sure something amazing: only finitely many of the coins came up heads. This is extremely unlikely—indeed, by the Law of Large Numbers it has zero probability—but it seems nonetheless possible. What probability should you now assign to your coin being heads?
Intuition: Very small, maybe zero, maybe infinitesimal.
Here's an argument, however, that you should stick to your guns and continue to assign 1/2. Let F be the proposition that only finitely many of the coins landed heads. Let G be the proposition that of the coins other than yours, only finitely many of the coins landed heads. Learning G does not affect your probability that your coin landed heads. The coins are all independent, so no information about the other tosses tells you about yours. But, now, necessarily (given the setup that you toss only one coin) F is true if and only if G is true. For your coin won't make the difference between infinitely and finitely many heads. So learning F does not affect your probability that your coin landed heads.
To make sticking to your guns even more amazing, note that this works for any infinity of people, even a very high uncountable infinity. Wow!