>> I think your opening "think about it" sums it up best. The answer is >> not obvious to me. I do personally hold some Platonic belief that each >> of the natural numbers, even all of those between 1 and googolplex, have >> an abstract existence. But I regard my viewpoint as more a statement of >> faith (perhaps something that is quote self evident unquote) than >> anything else. I think that questioning my viewpoint is a reasonable >> thing to do and not at all crankish. > > Isn't there also a simplicity argument here? The set of all natural > numbers, with no gaps and no end, is in many ways an easier thing to > reason about than the set of natural numbers that someone, somewhere, > has used or will use.
It makes some sense to talk about permuting the set of all natural number, say, by exchanging even and odd instances. With a finite set, it is meaningful to talk about a permutation which reverses the order of the set. What does it mean to reverse the order of the set of all natural numbers? What is the first half of the set of all natural numbers? Can the set of all natural numbers be evenly divided?
I have yet to fully accept the notion that it is meaningful (or useful) to describe the set of rational numbers as countable. I can follow the inductive argument which shows that the rational numbers can be recursively enumerated. At any given point we can stop the process and observe that for each rational number there is a unique natural number assigned to it, but when we start comparing that set to the set of all real numbers, as Cantor did, have we extended our abstractions too far?
The concept that the set of all natural numbers is infinite is certainly easier for me to grasp than the notion that it could somehow be finite. OTOH, when people start talking about the set of permutations of the natural numbers, I have less confidence that this is meaningful. -- Nil conscire sibi