Page 5 - Boundedness Revisited

shaved by the barber, and those shaving themselves and not
shaved by the barber; the barber is excluded from both groups
and therefore is not in the town; however, he must be in the town
to distinguish the two groups.” In both cases the paradox is
resolved by identifying an invalid unclosed inside.

b. Irrational insides
The fluctuations of this type of invalid inside result from
definitions based on indefinitely recurring functions. Each
occurrence or calculation of the function gives a new size of the
inside, which is said to approach a limit (which it cannot reach,
and thus to which it cannot be equivalent). As expressed by a ratio
of two whole numbers, an irrational “number” is incomplete; it is
therefore an invalid unclosed inside. Examples are 1/3, the square
root of 2, and pi. The relationship of boundedness to number is
further considered below.

2. Nonfictional boundaries

The aspects of boundedness denied by boundaries defined as
nonfictional are the continuity and arbitrariness of insides and
their outsides. Since one aspect cannot be denied without the
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other, they may be considered together. On a mundane level, the
use of nonfictional boundaries may be called simple reification.
Mechanisms of perception lead to the isolation or disconnection
of insides from their outsides; the feeling that such distinctions are
“objective” or not purely a function of perception does not stand
up to the reduction accomplished by boundary analysis.
On a more grandiose level, the intentional denial of continuity
found in scientific and religious theories creates what may be
called absolute nonfictional boundaries. These definitions reduce
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to “X is outside Y but is not continuous with Y.” In such cases,
analysis invalidates the inside or outside as well as the boundary

8 That is, their denial reduces to the same nonfictionality of a purported
boundary. Discontinuity means separation; ergo inside and outside have a real
boundary or a non-excluded middle. Non-arbitrary means real or objective,
ergo an end to X absolutely discontinuous with Y.

9 See footnote 2, in reference to macroscopic and microscopic limits.

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