... Either there is 'something' which has non-zero spatial extent occupying the exact central position of that real PIE disc object, OR there is an 'void' there that has non-zero spatial extent.
No your intuition is wrong.
Math is logic, but it must start with some assumed or defined objects. A "point" is one of these that by definition (an axiom) has no size, but has a location. Thus if you have a line segment between point A and B which are only 1 micron apart, there are an infinite number of points, all different from each other by their locations, on that 1 micron long line segment. Even more startling is fact that the number of points between A' & B' is an infinity just as large (and not bigger) as between A & B, even if A' & B' were a trillion light years apart.*
You "common sense" that there must be more points on line segment A'B' than on the 1 micron long AB, is wrong as this can be easily proven. Basically this proof goes like this:
Given any point on AB, P, there is one and only one point, P',on A'B' that correspond exactly to P (and conversely). For a specific example, consider the point x on AB, which is closer to A than to B by the fraction 41/ 333 or 0.123123123123123123.... there is the point x' (the only one) that is also 0.123123123123123123.... closer to A' than to B'.
As far as to which of the 1/3 slices of pie the center point "belongs to" the choice is arbitrary. If initially, when I cut conceptually the pie, it belonged to piece #2 and someone say: "No, it should belong to pieces #1, as that was the first cut." Then I say OK, lets move it to be part of piece #1.
Do you think after the move that the area of piece #2 is smaller and that of piece #1 has increased?
If yes, then you are illogical, or forgot the axiom defining what points are.
*likewise, there are no more points inside a cube one light year on an edge than on the 1 micon long line AB. Here is a nearly completed hint for you to proof that:
Consider the point inside the cube with Cartesian coordinates (a,b,c) and note that a, b, & c are expressed as decimal fractions of one light year. I.e a is a string of numbers 0 thru 9 like 0.2640684772456 without end usually but could be exactly 0.2 for the infinity of points in the bc plain with coordinates (0.2, b, c). In general a = a1 a2 a3 a4 etc., where for any point in the cube with a= 0.2640684772456 then for it, a1 =2, the a2 =6, the a3 = 4, the a4 =0 etc. for b & c.
Now on the 1 micron long line corresponding to point in the cube at (a,b,c) the is one, and only one, point with x-axis coordinates = 0.a1 b1 c1 a2 b2 c2 a3 b3 c3, a4 b4 .... etc.
And of course for any point on the 1 micron line there is one and only one corresponding point inside the light cube.
If there is an apple for every orange (one and only one corresponding to it) and an orange for every apple, then there are no more oranges than apples and no more apples than oranges - even your intuition should tell you that. But some infinites are bigger than others, but I won't detour more to prove that.
