I don't think I'm getting it. I admit the first dot will have infinity on its left, and infinity on its right. But as soon as you place a second dot, there must be a set distance between it and the first dot. Both dots will still have infinity on the side that faces away from the other dot, but the question "how far away is the first dot from the second dot" must, I think, have an answer.

Is this a well-known paradox, or did you come up with it? So I can maybe study this a bit.