Tl;dr: Could someone give me an explanation as to how infinitesimal numbers might be added together to get 1?
Why I want to know: some people (philosophers) like to introduce infinitesimals into probability theory when considering cases like an infinite amount of coin tosses. To say that the event that this coin will land heads every time has probability 0, they think, is bad, as probability 0 should be reserved for the impossible. Instead, they say it should have an infinitesimal probability.
Fair enough. But presumably, the probability that the coin corresponds to some infinite sequence of heads and tails is 1. If we like countable additivity, this means the probability of all the possible infinite sequences should sum up to 1.
Apparently this is consistent with their theory. Each infinitesimal probability of each infinite sequence will, when added together, equal 1.
I'm confused as as to how this could happen. Why would an infinite amount of infinitesimals add up to 1, and not, say, 2?