An interesting empirical example of the axiom of completeness is the night sky with a telescope of ever-increasing magnifying power. Take any space of darkness in the night sky and assume you can magnify as much as you want. The axiom of completeness asserts that you will eventually find a star in that space. Take another, smaller space within (not containing a star) and magnify more, you will find another star in that smaller space, or any space, no matter how small*.This, of course, is impossible under the standard physics mandate that there is an edge of the universe and it is not unlimited. In any case the idea of a limited universe is in direct tension with an elementary empirical (if we could inductively continue to magnify) example of the axiom of completeness. The use of stars instead of points gives an alternative to formulating analysis with 0-dimensional objects such as points. They appear like points only at certain levels of magnification.
Another iteration of the axiom of completeness is one in two or three dimensional space. The usual axiom expressed in two dimensions uses objects of 0-dimension: points, and asserts that a bounded increasing sequence (of points) has a least upper bound. Using more realistic objects of the sequence— instead of points, three dimensional shapes such as spheres or cubes—The cubes have to get smaller and smaller, and be contained in the previous cubes of the sequence (after cube N). The problem is that this sequence always contains some space, and asserts that the sequence converges to a point instead of a cube. This is not the inductive inference. The inductive inference requires that all cases that can be reasonably checked by hand resemble the cases beyond, approaching infinity. If the cases known resemble the cases beyond, there would always be some space inside the cube, and the axiom of completeness fails. All this is related to the unrealistic belief in 0-dimensional objects.
- just for fun lets be precise and say the next smaller circle has 1/4 the radius of the current circle. It is easy to see that this circle can always be found so that it does not include the star you found. Oh, and what if you find two (or more) stars at the same time? leave as an exercise!