Quote:
Originally Posted by Ether
You are using the word "uncountably" incorrectly here.
The integers are countably infinite.
The reals are uncountably infinite.
|
"Countably infinite" is still uncountable for us mortals. When you finish, please get back to me.
Quote:
Originally Posted by Ether
... the concept of half of an infinite set is not well-defined. It’s possible to pair up the even integers with the odd integers with none left over in either set, and if we were talking about finite sets, that would be a demonstration that each was half of the whole set of integers. However, it’s also possible to pair up the multiples of 100, say, with all the rest of the integers with none left over in either set, and the multiples of 100 are obviously only part of the set of even numbers. Clearly, then, this kind of pairing argument cannot lead to any very useful notion of half of the set of integers.
There is a notion of asymptotic density of a set of positive integers that does a pretty good job of capturing many people’s intuitive sense of what half (or any other fraction) of the set of positive integers should mean.
excerpted from http://math.stackexchange.com/questi...ll-numbers-odd
|
I fully admit to having (yes, consciously and deliberately) blown through the difference between aleph null and aleph one in my earlier reply, to make a statement that might make sense to shose who found the original problem "meaningless". (That is, I was more worried about enlightenment than mathematical rigor.) I specifically addressed your concern that the even integers and all integers are of the same cardinality.
A far as I am aware, any argument that decides the statement "Half of the integers are even." as
meaningless can also be used to decide the concept of "average length of a line segment located within the unit square" as meaningless -- or worse.
Edit:
Quote:
Originally Posted by Ether, as edited after I prepared the response above
It's meaningful only if you define what you mean,
If you mean "the set of all even integers has an asymptotic density of ½", then yes, it's meaningful. Otherwise, not.
|
OK, I'll go with that -- and it's a remarkably fast asymptotic function, as of every pair of consecutive integers, exactly one is even.
Here's a bit more precise statement: For every set of consecutive integers with a non-zero, even number of members, exactly half are even.