The Zero Delusion v0.99x

cannot retain the order relation because, given a ∈ Ň, neither a > ⊚ nor ⊚ > a
holds. For example, 2 ≯ ⊚ holds because 2 ≠ q · ⊚ + r. Likewise, ⊚ ≯ 2 because
⊚ = ⊚ · 2 + r = 2 · ⊚ + r but ⊚ ≠ ⊚, and ⊚ = q · 2 + ⊚ but 2 ≯ ⊚. Therefore, we
cannot identify ⊚ with ∞, let alone with 0. The indeterminate element is simply
an inaccessible natural.
Well, zero is not critical for N. What about the integers? The eagerness to
unite the negative and positive numbers was the chief reason to include zero,
making Z democratic so that any point can be an objective reference frame or
neutral coding source. Zero, "based on the dichotomy of source-evolution (origin
and derivate), has much to do with zero as a number between negative and
positive numbers" [111]. Here, evolution is progression implemented as a recursive
composition so that zero is the generator of positive and negative integers;
−→ S i (0) = −→ S i−1 (S (0)) = · · · = −→ S 2 (i − 2) = −→ S (i − 1) = i and ←− S i (0) = −i
for all i ∈ Z are the successor and predecessor operators. We agree that these
operators are vital, but zero is unnecessary to generate induction.
Above everything, additive groups include zero to bridge the requirement to
assign to each element in the group x another element y such that x + y = 0.
However, we object that a number canceling its opposite is a vacuous expression;
x + y vanishes. We can equivalently write it as y = −x showing that zero is
superfluous and exceptional because it is the only integer with no inverse. On
the contrary, if we accept that zero opposes itself, we might as well say that it
disobeys the axiom of additive inversion. Besides, the sum 0 + x = x and the
multiplication 0 |x| = 0 have as much sense as ∞ + x = ∞ and ∞ |x| = ∞.
The glamour of zero fools us deviating our attention. The power of fields,
rings, and algebras primarily resides in the multiplicative group, focusing on
invertible members, i.e., all nonzero elements. If needed, a field can represent the
void by adding the multiplicative unit and its opposite. Consider the alternative
definition of a field without zero in mathematics as a set equipped with the unary
operation of inversion, the binary operations of addition and multiplication, and
two constants, +1 and −1, so that it holds −x ≡ (−1) x and the expression
1 + (−1) ≡ 1 − 1 vanishes. We can embrace the Peano axioms for the definition
of the negative and positive integers, except that the initial case in induction
proofs requires using both units as generators instead of zero, i.e., −→ S n (1) = n+1,
←− S n (n + 1) = 1, −→ S n (− (n + 1)) = −1, and ←− S n (−1) = − (n + 1) for all n ∈ Ň,
while −→ S (−1) and ←− S (1) vanish. Consequently, zero is axiomatically unnecessary
for the integers.
Although the integers do not form a field, we can imagine a sort of "integer
projective line" that splits the indetermination ⊚ into 0 and ∞; the integer
line together with an idealized point at infinity, an antipodal point of zero that
connects to both ends of Z, traces a closed loop. In this set of extended integers,
zero enables a germinal version of the principle of general invariance, by which
fundamental physical laws must be coordinate-independent, converting zero into
a factotum, an almighty angle. Nonetheless, this covariance reinforces the need
for an arbitrary origin of coordinates and not the consideration of zero as an
integer.