The set N, that is the set of all positive integers, has infinite number of numbers, that is the number of numbers in set N is ∞, but none of the numbers in set N is ∞. As we remember, ∞ is not a number, especially it’s not an integer in the case of set N.
Let us assume that we start from finite number of people, but all of the people are immortal and at finite rate the number of people increases and that there’s endlessly time.
Can we even ask, that what is the number of people in these conditions ”eventually”? There’s no end, but we can’t just say that eventually the number of people would be infinite. Or can we?
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In the natural numbers set, there’s always a greater number, there’s no limit – but as said, none of the numbers is ∞. So would this be the case in the number of people in immortality – it would always get bigger and would never reach an end and never be ∞.
If the number of people would eventually be ∞, when would it be ∞ for ”the first time”? If the number of people would eventually be ∞, there would be a moment when the number of people would have a start but not an end; before that moment the number of people had a start and an end.
According to a joke, when asked when infinity is reached, a mathematician says ”never”, a physicist says ”sometime”, an engineer would say ”very soon”. 🙂
Well, this is just humor, the set theory is not model of reality.