Thinking about infinity

[Alex K]

"infinity" is not really a technical term for any particular (cardinal) number. If we are talking sizes of sets, both the natural and the real numbers have "infinite" size, but they are different infinities. The latter is larger than the former.

I wouldn't say larger, more like denser.
Like when I think about comparing infinites, I cut up both the infinites equally. Like,set of natural numbers cut up at 5. And the set of real numbers cut up at 5. So when we look at it the cut up version of the real numbers will have more numbers than the cut up version of natural numbers.
Like the cut up version of natural numbers will only have 1,2,3,4,5 whereas the cut up version of real numbers will have 1,2,3,4,5 and ever other number between them. So I think all infinities are equal in size but different in density. Like according to my example set of real numbers and natural numbers would be equal in size but different in density, like the set of real numbers would be infinitely denser than set of natural numbers.
Does that make any sense?  Lol

It makes sense, but that's a dangerous thing, because it is also wrong

The way size comparisons of sets are usually defined in mathematics is as follows:

Imagine you have two infinite sets A and B. If you can construct (or prove the existence of) a map which assigns each element of A to an element of B, covering all elements of A and B (a 1-to-1 map), it is said that the two infinite sets have the same size.

Now, consider the real numbers R and the natural numbers N. It can be proven that there is a map assigning each element of N to an element of R. One would then say that R is at least as big as N. Now, conversely, it can be proven that there is no map assigning each element of R to an element of N while covering all of R, i.e. the elements of R cannot, in principle, be enumerated by the elements of N. This is why, using the common definitions, the infinite set R is larger than the infinite set N.

[ErGingerbreadMandude]

I wouldn't say larger, more like denser.
Like when I think about comparing infinites, I cut up both the infinites equally. Like,set of natural numbers cut up at 5. And the set of real numbers cut up at 5. So when we look at it the cut up version of the real numbers will have more numbers than the cut up version of natural numbers.
Like the cut up version of natural numbers will only have 1,2,3,4,5 whereas the cut up version of real numbers will have 1,2,3,4,5 and ever other number between them. So I think all infinities are equal in size but different in density. Like according to my example set of real numbers and natural numbers would be equal in size but different in density, like the set of real numbers would be infinitely denser than set of natural numbers.
Does that make any sense?  Lol

It makes sense, but that's a dangerous thing, because it is also wrong

The way size comparisons of sets are usually defined in mathematics is as follows:

Imagine you have two infinite sets A and B. If you can construct (or prove the existence of) a map which assigns each element of A to an element of B, covering all elements of A and B (a 1-to-1 map), it is said that the two infinite sets have the same size.

Now, consider the real numbers R and the natural numbers N. It can be proven that there is a map assigning each element of N to an element of R. One would then say that R is at least as big as N. Now, conversely, it can be proven that there is no map assigning each element of R to an element of N while covering all of R, i.e. the elements of R cannot, in principle, be enumerated by the elements of N. This is why, using the common definitions, the infinite set R is larger than the infinite set N.

Wow, that makes 1000 times more sense. Thanks!