Add your own or talk about those under discussion.
To start off with, I have a couple of strange thoughts about infinity. I'm not sure how original they are.
Firstly, a metaphysical problem: infinite lifespans. We don't have much trouble with the concept of immortals; they are born like us, and imagining what it would be like to be one is just a matter of extrapolating from our current life experience. But what about the other direction? What about an immortal being which, additionally, has always existed? Meeting and conversing with such a being seems plausible. But what about being that being? Intuitively it seems like it's 'impossible for its consciousness to ever reach this point in time'. Why is this so different from the classic conception of an immortal? What about objects in general? Could you encounter a rock which had existed forever? Or a clock?
Secondly, a moral problem: infinite beings. We think killing is immoral in what we presume to be a finite universe. But what if we discovered that the universe was infinite? Would it be moral to destroy a planet with life on it? If you did so, you would not have reduced the amount of life in the universe at all. This is not controversial; refer to the Hilbert's Hotel analogy if you're confused. You could just find another planet with life on it (in fact you could find an identical planet) and put it where the old one was, and then replace the new empty space with another planet, and so on with different planets forever, leaving you with exactly the same arrangement as before you destroyed the planet.
Developing the tangent about there being infinite identical planets: to what extent is it correct to say that they are individual? Would there be multiple copies of your consciousness? Or just the one? How is this any different from encountering a mirror?
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