Thinking about infinity

[Excited Penguin]

Some sets of infinities certainly seem logically possible. 

For example, the infinite set of natural real numbers seems to exist, which is to say that there is an infinity of natural real numbers existing even now, and there is none of them which do not yet exist.

There are an infinity of points on a line segment.

It is logically possible that reality has an infinite history of cause and effect (i.e. extending "backward" through the big-bang), and that it will have an infinite future.

In what sense could an infinity of simultaneously existing things exist together as a finite thing?

The set of natural real numbers aren't really infinite, we only think of them as such. Same holds for the points on a line segment. In the real world, nothing has been proven to be infinite, nor could it be. Feel free to dream of it though.

In my uninformed, uneducated opinion infinity is illogical and doesn't have a place in the real world.

Do correct my ignorance.