The Zero Delusion v0.99x

the most significant proof of the intimate fellowship between zero and infinity
comes from logic. According to Gödel’s theorems [5], we can find unprovable
statements in a contradiction-free and enumerable logical system; vice versa, if
such a system is complete (the truth or falsity of any axiomatically-constructed
piece of knowledge is provable within the system), we can find incongruencies.
Thus, a system requires null or uncountably infinite cardinality to reach axiomatical
completeness and consistency simultaneously; logical insufficiency and
uncomputability are crucial properties that unite zero and infinity.
In mathematical physics, the Dirac delta "function" [41] is a prime object
that masters the zero-infinity duality. It serves as a distribution gateway to the
continuum, and vice versa, from the reals to discontinuous math. Its output
vanishes everywhere except when the domain value is zero, whereas its integral
over R equals one. This function is also of paramount importance in physics
because it allows calculating a system’s response to the input’s total, i.e., as
an impulse (e.g., an instantaneous collision) or potential (e.g., a point mass),
escaping from the transition processes or layers at lower integrative levels.
Is the Dirac delta distribution physical? No, it is only the abstraction of
an immeasurable perturbation extended over no period. In subsection 2.2, we
will justify that a being’s existence requires taking up nonzero spacetime, and a
change needs nonzero spacetime to perform. Even algebraically, we must reject
the utter punctuality of zero and infinity; "with the extension of variable magnitudes
to the infinitely small and infinitely large, mathematics, usually so strictly
ethical, fell from grace" [43]. We understand the Dirac delta better as the limit
rendered by a finite sequence of distributions δ α (x) = 2 /( √ π|α| exp(( x /α) 2 )), with
α ∈ ˇQ. As α dissolves, the bumps get sharper and sharper, concentrating on
a prong or pin at the origin. Thus, we recover the mathematical and physical
sense of this duality of beables operating as a logarithmic scale’s communicating
vessels; δ α→log 1 (log 1) ∝ 1 /|α| diverges, whereas δ α→log ∞ (log 1) ∝ 1 /|α| vanishes.
Besides the Dirac delta distribution, we can find in other branches of mathematics,
such as projective geometry, and physics, such as cosmology, examples of
the symbiotic bond between the prospect, not the actuality, of zero and infinity
providing us with a realistic vision of the universe.
Combining both concepts leads to formidable mathematical tools and physical
models. Since {0, ∞} /∈ Ǎ, the ratio z /α exists ∀ {α, z} ∈ Ǎ;
z/α
lim
|α|→log 1
diverges, and lim z/α vanishes. Similarly, line curvatures in the Ǎ plane are
|α|→log ∞
tacitly nonzero; specifically, rays are not straight. Ǎ interprets infinity as "the
last point" observed from the origin at the end of every line, wrapping around
a giant circle, where zero and infinity are antipodal points (180 degrees apart).
This perspective allows us to identify the zeroes and poles of a modular or
Möbius transformation with limiting values like 0 and ∞, around which we can
define an arbitrary open neighborhood where the map regularly behaves without
exception. Indeed, the modular group’s action is the automorphism group of the
"real-algebraic projective line" (mapping real-algebraic numbers to themselves