random thought: are some infinite sets larger than other

Question

I was in the shower today and I just thought of this so I'm asking it. I'm sure this has been thought of before.

Let's say we have two sets, the set of all even numbers and the set of all natural numbers. They are both infinite, right?

But let's say we cut off the set at any number $n$. So for example if $n = 4$ then the even set would be ${2,4}$ and the natural set would be ${1,2,3,4}$.

So the set of natural numbers is bigger when they both reach 4, since they both increase in the same direction in a linear fashion that doesn't seem to be an unreasonable comparison to me. So if we take $n = \infty$ , then why are both sets the same size? In other words, is it possible that one infinity is greater than another?

I'm sure this is not the case, but can someone please explain? Thanks

Answer 1

It is certainly possible that one infinity is greater than another. With sets, we generally talk about size in terms of bijections: two sets are the same size if they can be put in bijection with each other (A bijection is a map from one set to the other such that everything is the second set is mapped to exactly once). The two sets you describe have a bijection: we send every even number to half of itself. So in set-theoretic terms, they have the same size. However, there are sets that are strictly larger: what are known as the real numbers CANNOT be put in bijection with the natural numbers. This was proven by Cantor with his diagonalization argument.

The concept you're looking for is known as the natural density. This is a way to count just how much smaller than the natural numbers a given subset is: if, roughly, it has one number for every $a$ natural numbers, then it has natural density $\frac 1a$. Your set is a particularly nice example: it has natural density $\frac 12$ because it has exactly one number for every two natural numbers! At the same time, there are many important sets that have natural density $0$ (in other words, there are far fewer of them than natural numbers in total): the square numbers, for insance, or the prime numbers.

Answer 2

I think you're confused about what a real number is. There are infinitely many real numbers less than $4$, including $\frac{1}{2}$, $\pi$, $\sqrt{2}$, etc. Instead of the set of real numbers, you seem to be talking about the set of natural numbers.

But putting that aside, you seem to be talking about the concept of Asymptotic Density for subsets of the natural numbers.

Answer 3

The short answer is yes, it is possible that one infinity is greater than another. You need to clearly define what this means. In the usual definition, Cantor's diagonal argument shows there are more reals than integers. The long answer is that the subject is (almost) all of set theory.

Answer 4

Yes, you've touched on something that is indeed true and interesting. Some infinities are larger than others, and this is a well-discussed topic on M.SE.

See, for example: different kinds of infinities
are all infinities equal?
You may also find this article on Aleph Numbers informative. Good Luck! I'm voting to close this question as it is fundamentally a duplicate.