The organisation of thought, educational and scientificWhitehead, Alfred North
Philosophy
The organisation of thought, educational and scientific
Whitehead, Alfred North
Education
Finally, by keeping only the central point fixed, and by making _h_,
_b_, and _c_ diminish indefinitely, the series converges to a point.
Furthermore, we have introduced as yet no concept which would prevent
an enclosure-object being formed of detached fragments in space. Thus
we can easily imagine a convergent set which converges to a number of
points in space. For example, each object of the set might be formed of
two not overlapping spheres of radius _r_, with centres _A_ and _B_.
Then by diminishing _r_ indefinitely, and keeping _A_ and _B_ fixed, we
have convergence to the pair of points _A_ and _B_.
It remains now to consider how those convergent sets which converge to
a single point can be discriminated from all the other types of such
sets, merely by utilising concepts founded on the relation of enclosure.
Let us name convergent sets by Greek letters; by proceeding "forward"
along any such set let us understand the process of continually passing
from the larger to the smaller enclosure-objects which form the set.
The convergent set α will be said to "cover" the convergent set β, if
every member of α encloses some members of β. We notice that if an
enclosure-object _x_ encloses any member (_y_) of β, then every member
of the "tail-end" of β, found by proceeding forward along β from _y_,
must be enclosed by _x_. Thus if α covers β, every member of α encloses
every member of the tail-end of β, starting from the largest member of
β which is enclosed by that member of α.
It is possible for each of two convergent sets to cover the other. For
example, let one set (α) be a set of concentric spheres converging to
their centre _A_, and the other set (β) be a set of concentric
cubes, similarly situated, converging to the same centre _A_. Then
α and β will each cover the other.
Let two convergent sets which are such that each covers the other be
called "equal."
Then it is a sufficient condition to secure that a convergent set
α possesses the point type of convergence, if every convergent set
covered by it is also equal to it, namely, α is a convergent set with
the punctual type of convergence, if "α covers β" always implies that β
covers α.
It can easily be seen by simple examples that the other types of
convergence to surfaces or lines or sets of points cannot possess this
property. Consider, for example, the three convergent sets of boxes in
the preceding illustration, which converge respectively to a central
plane, a central line in the central plane, and the central point in
the central line. The first set covers the second and third sets, and
the second set covers the third set, but no two of the sets are equal.
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