Double-infinity

This is a strange thing for even me to envision. I have had a nagging thought about what exactly infinity is. I just subconciously thought about it in eighth grade and forgot, then remembered today and decided to explore.

In geometry, I remember when I wondered if the Ray was infinite in length. It seemed so, but the problem was that it had a beginning. I wondered then what the line was. It was, well, two rays back to back. Infinity * 2, and that got me thinking if perhaps there is a separate infinity.

Picture a coordinate plane, with an inequality of Y > X. Half of the plane is shaded in, and it becomes infinite in solutions. But it has another half of, as well, infinite non-solutions. How can this be on the same plane? I always thought that there was an extra infinity, making there exist two infinities on the same plane. So, I figure that there has to be a double infinity.

This was kinda bruised when I graphed a Y > X^2, because the inner parabola was infinite in solutions, and the outer one as well, but they weren't half-planes, because it wasn't a line.

Perhaps this isn't a new idea, just one I haven't heard elsewhere. Any thoughts?