Correspondence with Graham Oppy on mathematical difference between infinite & indefinitely long lifespans; part 3

For the initial email exchange please see: Correspondence with Graham Oppy on mathematical difference between infinite & indefinitely long lifespans; part 1

For second email exchange please see: Correspondence with Graham Oppy on mathematical difference between infinite & indefinitely long lifespans; part 2

Email #5, 24th of February, 2020

Hi Graham,

the difference between indefinite and infinite lifespan I want to grab mathematically is this, conceptually framing it: if one is treated with what I call Open Healthspan Technology, all of the known internal ageing related caused of death is prevented, but this does not mean that there won’t be unknown ones emerging that might kill people off, and it certainly does not mean any external causes of death (all of your wipe-our scenarios and much more) will be eliminated, so this means indefinite lifespan with a daily non-zero mortality rate.

As opposed to this scenario, infinite lifespan is what I call immortality means a zero mortality rate on a daily basis, so practical invincibility that applies to all known ways to death (internal and external) that now can be avoided. And here we can include all of your wipe-our scenarios too. This is the only conceptual opportunity that I see can guarantee immortality. And this is the scenario that is utterly impossible from a biomedical, carbon-based human point of view.

So I want a mathematical analogy that makes these two scenarios easily or at least accurately distinguishable.

Does my question make more sense now?
Best,
Attila

Email #6, 26th of February, 2020

Hi Attila!
I think you have already provided a mathematical specification.
If you are immortal, then, on any given day, the probability that you will die on that day is exactly 0%. 
If you are immune to known internal — ageing related — causes of death, you are still vulnerable to external causes of death, and to hitherto unknown internal — ageing related — causes of death. While we cannot give an exact number, we can say that there is some epsilon such that, on any given day, the probability that you will die — from external or unknown causes — on that day is epsilon.
What is the probability that you will be alive N days from now? I think: (1-epsilon)^N. While this value tends to 0 as N tends to infinity, it has some finite value for each N. Moreover, if epsilon is very, very small, this condition is only significantly discriminable from immortality in the very, very long term.
Cheers,

Graham