Given any two categories, there exists at least three and no more than four possible categorical relationships between them:
|No members of one are members of the other.||Some members of each are contained, and not contained in the other.||One is contained|
in the other.
If you want to count two identical categories as two categories, then "identical" is a fourth possible categorical relationship. You can also treat identical categories as one category. It usually depends on whether they are definitionally identical or coincidentally identical.
These categories are always identical.
These categories are identical relative to a particular place at a particular time, but not necessarily always identical.
There are at least three kinds of hierarchical relationships. The contained set is either a subset, a part, or a subordinate of the container set. e.g.
Engines and tires are parts of cars.
Lieutenants and sergeants are subordinates of captains.
Sets may be categorized in at least three ways: e.g.
A null set contains no concepts.
a. by the number of criteria their members have in common
Composite sets have more than one criterion in common
Simple sets have one type of criteria in common
Complex sets have more than one type of criteria in common
Distinct sets have clear boundaries
Vague sets have at least one unclear boundary.
Concepts may be categorized by boundary.
Distinct concepts have clear boundaries.
Vague concepts have at least one unclear boundary.
e.g. zero, emptiness, vacuum, nothing, absence, point
e.g. Space is infinite in three dimensions (assuming either an infinite universe, or an infinite number of universes in one space continuum)
A plane is infinite in only two dimensions.
A line is infinite in only one dimension, but two directions.
A direction is finite on one end and infinite on the other.
e.g. everything, anything, existence
Paradoxically however, infinity becomes a limit in itself.