Hi Yanniru,
"*I also do not understand how to scale an infinite system.*"
I don't know exactly what you mean by "to scale". Normally, it means to change the size by multiplying all linear measurements by a constant. And, as you must know, when you scale something, not everything gets scaled the same way. That is why a grasshopper can jump many times his body length while a hippo cannot. But, if you only consider linear measurements, and not volume or other quantities, then a scale model airplane is a reasonable facsimile of a real airplane.
Now, if we try to apply that notion of scaling to an infinite system, it doesn't work. The cardinality of an infinite system is not a number so there is no reasonable definition of multiplication for it. As you know, there are many (some say an infinite number of) orders, or cardinalities, of infinite sets, but to claim that these represent different scalings is to distort the concept of scale beyond meaning, IMHO.
I am of the firm opinion that any consideration of infinities leads to nonsense. Cantor's method led to paradoxes immediately, and in my view they should have been sufficient cause to reject the notion outright. Instead, mathematicians have pussyfooted around the antinomies and accepted the nonsense into their systems. I think that Goedel's Theorem should have been seen as strong supporting evidence for my position since it screams out that you can't have a consistent, complete, infinite system. But, alas, people keep their infinities anyway even though I know of no useful purpose they provide any branch of mathematics. And, they definitely get in the way of science's use of mathematics. (Boy, that poor old dead horse sure gets a beating from me!)
"*It would seem that incompleteness is then never realized in nature. Is that so?*"
Good question. I think it depends on what you mean by "complete". If you mean 'all there is', then nature must always be complete. But if you mean 'all there could be', then I think nature is always incomplete. But in no case (watch out dead horse) do I think anything in nature can be infinite.
"*I have been doing arithemtic most of my life without ever needing to consider infinity.*"
So has everybody else. No computation ever performed by any person or by any machine ever needed or used infinity and none ever will.
"*Is the finite system of arithematic I am using therefore complete?*"
No. Mathematically speaking it is incomplete. But, including infinity doesn't necessarily help. For example, the infinite set of integers is incomplete because it does not contain the answers to all division questions. For example, 5 divided by 4 does not appear in the infinite set of integers.
Warm regards,
Paul |