A "point" doesn't require intersecting segments to exist. If there are intersecting segments, a "point" is recognizable and may be quantified, labeled and graphed.
If one postulates 2 dimensions must exist for a single dimension to exist and express, and 3 dimensions must exist for a 2 dimensional segment to exist and express and so on, the net effect is going to be "infinite dimensions", each being a necessity to express the last. Ultimately nothing would be quantifiable because those "infinite dimensions" form a singularity, or a point.
Euclidean space is not infinite and expressly so by the fact of correspondence. You've thought yourself into a circle trying to redefine parameters with a sort of "word game", but the point isn't entirely lost when considering the potential for "additional, relatively unperceived vectors".
Instead of attributing the values "all" or "always" (is) or "least common" (is shared), or mislabeling hypothetical additional vectors as "4th dimensions" and so on, since you're interested in this, you might consider investigating others' learned views on the matter. You'll need a strong background in math and alot of time.