On a side-note, why are you arguing with me on this?

You came up with that, adressing the problem, if earth can be an infinite plane, or if it is philosophically (mathematically?) impossible. I just question your terminology.

Perhaps I should have initially modified "infinity" with "actual" so as to address the distinction before you brought it up. There are philosophers of mathematics on both sides of the issue, who believe an actual infinity can't exist (for example, Gauss believed that we only approach infinity, this is the view I hold; more recently, the philosopher of religion William Lane Craig has argued for the finitude of the Universe based on Hilbert's paradox) and there are those who believe it can exist, often for theistic reasons, to apply it to the Divine.

For example: a plane

contains a finite number of discrete elements, with a continuous infinity in between each discrete element.

I never heard that a plane contains a discrete number of elements. With some infinity between them. What are these discrete elements?

This time, I'll attempt to explain using interval notation. Say we have a distance of 3 units. We'd write this mathematically as [0,3]. In this way, we have a distance with a potential infinity -- the number of points in this distance

*approaches* infinity, but it is in a sense "contained" within that 3 miles. This is the reason we can traverse finite distances -- we aren't actually traversing an infinit e number of points.

Now take an infinite distance. We'd write this mathematically as [0,∞). In this way, an actual infinity is instantiated. It isn't merely approaching infinity -- there are actually an infinite amount of units in this distance. It may be hard to grasp, but think of it conceptually, in terms of distances.

Have you ever heard of the paradox of Achilles and the Turtle? Aristotle's solution to the paradox was to distinguish between these two infinities -- one that is finite, merely approaching an infinity within the certain distance (i.e. [0,3]) and one that is actually infinite, instantiating the infinity as its distance (i.e. [0,∞)). It's an important distinction to make. Hilbert's Hotel is a paradox that applies to the latter, because it's not a finite "distance," it's an infinite "distance." Perhaps I was throwing you off with the "discrete" and "set" terminology.