The Zero Delusion v0.99x

use a negative base for negative numerals. We can even combine the power of
bijective notation with the efficiency for calculations of the signed-digit representation,
say the "non-adjacent form" [108] (or canonical signed digit representation)
or the classic "balanced ternary" system [75]. Let us focus on the latter,
the best number system regarding global computability [99].
Balanced ternary only uses the symbols {↓, 0, ↑}; for example, ↑ 0 ↑↓=
3 3 + 3 1 − 3 0 = 29 and ↓ 0 ↑↓= −3 3 + 3 1 − 3 0 = −25. This system proves that we
can split any integer into a positive and a negative number, a sort of double "binary"
system, e.g., −25 = −28+3 = [1001; 10] 3
or −25 = [0 . . . 01001; 0 . . . 010] 3
.
To dodge the prepended zeroes problem of this notation, we can write the negative
and positive components in bijective base-3 numeration, e.g., 29 =↑ 0 ↑↓=
[1; 233] 3
= −1 + 30 = −1 × 3 0 + 2 × 3 2 + 3 × 3 1 + 3 × 3 0 , 30 =↑ 0 ↑ 0 = [; 233] 3
,
and −25 =↓ 0 ↑↓= [231; 3] 3
= −28 + 3 = −2 × 3 2 − 3 × 3 1 − 1 × 3 0 + 3 × 3 0 .
With the convention that rationals delegate the "negative sign" to the numerator,
we can write the rational −25 /29 as the pair of pairs of bijective coordinates
(231; 3 / 1; 233) 3
and represent it as a unique rectangle of nonzero size in a twodimensional
grid Ž2 , where Ž is the set of nonzero integers (see subsection 1.1).
Thus, this "bijective balanced ternary" representation of rationals is a zero-free,
nonzero-size, unambiguous, and efficient numeral system.
Moreover, we can connect the basic code system denominated Canonical
Representation for the nonzero natural numbers with bijective numeration to
yield a zero-free prime-based factorized representation of the natural and rational
numbers. By the Fundamental Theorem of Arithmetic [143], we can think
of P, the set of all prime numbers, as the atoms of N so that we can express
every natural greater than one as a unique finite product of primes; for example,
16857179136 = 541×23×7 2 ×3 3 ×2 10 . The "arithmetic" of this prime factorization
consists of binary operations such as the product, greatest common divisor,
and least common multiple, whose outcome admits a representation again in
terms of the prime factors of the operands themselves. To banish zero, we can
express the prime subscripts and multiplicities as a pair of components in the
bijective notation; for example,
16857179136
29
= {(3131, 1) (31, −1) (23, 1) (11, 2) (2, 3) (1, 31)} 3
,
i.e., p 1 100×p −1
10 ×p1 9×p 2 4×p 3 2×p 10
1 , where p n is the n-th prime number.
4 Zero Countability
On how the theory of sets struggles to endorse collections of nothing and why
zero is axiomatically unnecessary.
4.1 Emptiness
ST is the branch of mathematics founded by Georg Cantor that analyzes groupings
of objects relevant to number theory, relational algebra, combinatorics, formal
semantics, and fuzzy logic. A set is a well-determined collection of things;