Page 9 - Boundedness Revisited

The foundation of logico-mathematical theorizing is the answer
to the question, “what can be counted?” The principle of
boundedness provides but one response: “any inside.” A number
is a count of valid insides; as shown above, boundary analysis
14
invalidates irrationals as numbers. Beyond this the principle of
boundedness is useful as a metalogical tool for criticism of the
theory of metamathematics originated by Cantor.

According to the Dedekind-Cantor axiom, any finite line
segment contains an infinity of points which correspond to an
infinite count, including irrationals. The violations of boundedness
are manifold in this axiom, beginning with points. A point is not,
or has not, an inside. Points, therefore, cannot be next to each
other or separate one from another. Although any inside can be
divided and subdivided indefinitely, each distinction is a fictional
boundary which does not begin to exhaust the nonfictional
content of that inside. It should also be remembered that insides
lacking known dimensions are not valid; lines, planes and timeless
solids are as uncountable as points.

The illogical outcome of point-counting is the notion of
transfinity: if between any two points an infinity of points
separately exist, then the count of any line segment “matches” that
of any other. The invalid boundary created for this matching is of
the absolute nonfictional type; its implication, as pointed out by
critics when Cantor first described it, is to deny the basic tenet of
logic (i.e., boundedness) that a whole (an inside) is greater than any
of its parts (lesser insides inside the first inside). When actual
numbers are used to match “different” infinite series (such as 1, 3,
5, 7, 9, … or 86, 87, 88, 89, …) violations of boundedness result
from the nonfictional boundaries necessary to leave out the
missing numbers.
Because of contemporary objections, the invalid assumptions
of Cantorian metamathematics were legitimized by an absolute

14 This should have stated that a finite count of numerals is a series of valid
insides. This entire section is rather presumptuous and polemical; it may be
ignored without affecting the rest of the essay. Its main purpose is to identify
the reification of invalid insides (nothing and infinity), a hazard only to non-
mathematicians wondering how such ideas map to reality.
8