| | Glenn wrote, Bill,
As usual, you have shown "infinite" patience. But as Robert sarcastically said: I just don't get it. Your arguments are clever, but so were Zeno's. I'll have to think about it some more. Nothing you have said has affected my intuition that it's possible that I can go in some direction and count the objects in the universe and keep going and keep counting and never reach the end. It's possible that for every integer there is a different object in the universe. Infinite sets can exist and the universe may be one of them. Part of the problem may stem from a confusion between abstract numbers and concrete objects. A number set isn't finite in the same way that a set of objects is finite. A set of numbers can go on infinitely, in the sense that however far you've counted, you can always count another number. But observe that in the process of counting another abstract number, you are mentally creating that number; you aren't finding it, as you would be if you were counting actual objects that you discover in the real world. In other words, the number doesn't already exist in the way that another object that you discover already exists. So there is no limit on how many abstract numbers you can count, because you are mentally creating them as you go, whereas there is a limit on how many objects you can discover: the limit is the number that actually exists in the real world.
What you are saying, when you say that the number of objects in the universe is infinite is that for every object in the universe, there is yet another object -- which is a contradiction, because since every object exhausts all the objects, there cannot be another object. The notion of an actual infinity of existents is, therefore, self-contradictory. In fact, you can't even imagine it.
What you can imagine is to find another object in addition to the objects you've already found and to continue doing so without reaching an end. But observe that to imagine not reaching an end is to imagine doing so within a finite period of time, which leaves open the possibility that if you were to continue the process, you would reach an end. In order truly to imagine the existence of an infinite number of objects in the universe, you would have to imagine counting them for an infinite period of time, which is impossible.
- Bill
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