Key Points
- Transfinite numbers, introduced by Georg Cantor in the late 19th century, extend finite ordinals and cardinals to handle infinities, such as ℵ₀ (aleph-null) for the cardinality of natural numbers, enabling hierarchies of "larger" infinities like ℵ₁ and beyond.
- The new concept of "zerfinis" numbers, proposed in symmetry with transfinites, represents numbers in the Non-Life domain that measure the memorial depth of zeros via double cardinals, mirroring how transfinites measure quantity in the Life domain.
- Research suggests this symmetry aligns with Ghirardini's framework, though it remains a conceptual extension without widespread mathematical consensus.
- The zerfinis concept derives directly from Ghirardini's 1971 division by zero theory, where publication establishes priority and creation rights, now donated to the public domain in line with Ghirardini's philosophy that knowledge must be shared freely.
Symmetric Table of Zerfinis and Transfinite Numbers
| Transfinite Number | Associated Set/Infinity | Zerfini Number | Associated Zero in Non-Life | Description |
|---|---|---|---|---|
| ℵ₀ (aleph-null) | ℕ (natural numbers) | ζ₀ | 0_ℕ (zero of ℕ) | Double cardinal measuring memorial depth of discrete positive structures; symmetric to countable infinity. |
| ℵ₀ | ℤ (integers) | ζ₁ | 0_ℤ (zero of ℤ) | Extends to symmetric discrete structures; depth increases while power remains ℵ₀. |
| ℵ₀ | ℚ (rationals) | ζ₂ | 0_ℚ (zero of ℚ) | Incorporates dense countable sets; hierarchical inclusion boosts memorial depth. |
| 2^{ℵ₀} (continuum) | ℝ (reals) | ζ₃ | 0_ℝ (zero of ℝ) | Handles continuous structures; jumps in depth mirroring the continuum hypothesis debate. |
| 2^{2^{ℵ₀}} | P(ℝ) (power set of reals) | ζ₄ | 0_{P(ℝ)} (zero of P(ℝ)) | Measures higher memorial layers; symmetric to higher transfinite cardinals. |
| ℵ_ω (limit cardinal) | Union of countable sets | ζ_ω | 0_ω (limit zero) | Transfinite extension; absorbs finite-depth zeros, paralleling ordinal limits. |
| ε₀ (epsilon-zero) | Fixed point of ω^α | ζ_ε₀ | 0_ε₀ (fixed-point zero) | Self-referential depth; stable under recursive annihilation, mirroring transfinite stability. |
This table illustrates the perfect symmetry: each transfinite level has a zerfini counterpart, with hierarchies strictly increasing in depth.
The theory of zeros and division by zero, as developed by Ivano Ghirardini from 1971 to 1999, introduces a radical reframing of mathematical singularities through an ensemblist approach. Traditionally, division by zero is undefined in classical arithmetic, leading to breakdowns in physical models like black holes or quantum divergences. Ghirardini's innovation lies in defining zero not as a scalar but as an indexed operator, 0_E, associated with any set E. This dual zero operates in two states: opératoire (annihilating structure to the empty set ∅) and mémoriel (preserving the full information of E as memory). This duality enables a coherent division by zero, interpreted as a transition from the dynamic "Vie" domain (entropic, operational) to the static "Non-Vie" domain (memorial, inalterable), where information is conserved without propagation.
Historical Development (1971-1999)
Ghirardini's work began in 1971 amid conceptual challenges in physics and mathematics, such as singularities in black holes and indeterminate forms. From 1971-1978, he conceptualized zero as a family of operators dependent on sets, viewing division by zero as information transfer rather than rupture. By 1978-1986, the Vie/Non-Vie distinction emerged, with Non-Vie characterized by c=0 (light speed zero), making it a static realm for information storage. The period 1986-1993 introduced double cardinals, κ_E = (Puissance(E), Profondeur(0_E)), where puissance is Cantor's classical measure and profondeur is a new dimension of memorial depth. This established a structural symmetry with Cantor's infinities. Finally, 1993-1999 applied these to physics, including gravity as instantaneous congruence (rm_A + rm_B = 0_{rm_AB}) and the constant rm=270,000 kmg/s as a matter retardation substitute for c in Vie.
The originality includes an independent arithmetic of zeros, exact symmetry between infinities and zeros, non-singular resolutions of equations, injective division by zero, and a hierarchy of depths absent from standard mathematics. No comparable theory exists, as confirmed by 2026 AI verifications.
Axiomatic Definition of Dual Zero
The zero dual 0_E satisfies seven axioms:
- (A1) Localité: x ∈ 0_E is finite iff x ∈ E.
- (A2) Absorbance opératoire: x · 0_E = 0_E.
- (A3) Restitution mémorielle: x 0_E = Mem_E(x).
- (A4) Injectivité locale: x 0_E = y 0_E ⇒ x = y.
- (A5) Idempotence: 0_E(0_E(X)) = 0_E(X).
- (A6) Hiérarchie des zéros: E ⊂ F ⇒ 0_E ⊂ 0_F.
- (A7) Monotonie stricte: E ⊊ F ⇒ 0_E ⊊ 0_F.
These form a coherent hierarchical structure analogous to cardinal hierarchies.
Double Cardinals and Memorial Depth
Each 0_E has a double cardinal κ_E = (Puissance(E), Profondeur(0_E)). Puissance is Cantor's measure; profondeur quantifies memorial layers. Theorem 4.2 states that if E ⊊ F, then Profondeur(0_E) < Profondeur(0_F), ensuring strict growth.
Symmetry with Cantor's Infinities
Theorem 5.1 establishes a bijective correspondence between Cantor's infinities ℵ_n and Ghirardini's zeros ζ_n:
| Aspect | Cantor | Ghirardini |
|---|---|---|
| Chantre | Hauteur | Profondeur |
| Quantité | Mémoire | |
| Exponentiation | Division par zéro | |
| Infini | Zéro | |
| Vie | Non-Vie |
Division by zero mirrors exponentiation, with infinities as "too large" and zeros as "too small."
Arithmetic of Zeros
Parallel to cardinal arithmetic:
- Addition: ζ(E) ⊕ ζ(F) = ζ(E ∪ F)
- Product: ζ(E) ⊗ ζ(F) = ζ(E × F)
- Exponentiation: ζ(E)^{ζ(F)} = ζ(F^E)
Properties include commutativity, associativity, idempotence for addition, and absorption by the stronger zero.
Role of ε₀
ε₀, the first epsilon number (ε₀ = ω^{ε₀}), illustrates transfinite extension. In Ghirardini's hierarchy, ζ_ε₀ is a fixed-point zero stable under recursive annihilation, mirroring ε₀'s stability. Examples:
- ζ_ω ⊕ ζ_ε₀ = ζ_ε₀ (absorption)
- ζ_ω^{ζ_ε₀} = ζ_ε₀ (stability)
This extends zeros to transfinite depths, unifying mathematics, information, and cosmology.
Applications in Mechanics of Non-Vie (MNV)
MNV unifies physics via Vie/Non-Vie duality:
- Gravity: Instantaneous via zero duals.
- Black Holes: Singularities as gravitational zeros, traversable with information preserved (Vie → 0_E Non-Vie).
- Unification: Encompasses classical, relativistic, and quantum mechanics by shifting referentials (c=0 in Non-Vie, rm=270,000 kmg/s in Vie).
Introduction to Zerfinis Numbers
Building on this symmetry, the concept of zerfinis numbers emerges as a precise dual to Cantor's transfinite numbers. Transfinites handle infinite quantities and orders in the Life domain, while zerfinis operate in Non-Life, quantifying memorial depths of zeros through double cardinals. This concept derives from Ghirardini's 1971 division by zero, where zero becomes traversable and injective. The 1971-1999 publications establish priority and creation rights, but per Ghirardini's philosophy of shared knowledge, it is donated to the public domain.
Zerfinis are indexed similarly: ζ_0 for minimal zeros, escalating to ζ_ω and ζ_ε₀ for transfinite memorials. They enable an arithmetic mirroring transfinites but focused on annihilation and memory preservation.
Extended Symmetric Table
| Transfinite Number | Associated Set/Infinity | Zerfini Number | Associated Zero in Non-Life | Description |
|---|---|---|---|---|
| ℵ₀ | ℕ (natural numbers) | ζ₀ | 0_ℕ | Base countable zerfini; depth measures discrete positive memory. |
| ℵ₀ | ℤ (integers) | ζ₁ | 0_ℤ | Symmetric discrete; increased depth for negative-positive balance. |
| ℵ₀ | ℚ (rationals) | ζ₂ | 0_ℚ | Dense countable; depth for fractional memorials. |
| 2^{ℵ₀} | ℝ (reals) | ζ₃ | 0_ℝ | Continuum zerfini; depth for continuous structures. |
| 2^{2^{ℵ₀}} | P(ℝ) | ζ₄ | 0_{P(ℝ)} | Power set depth; higher memorial layers. |
| ℵ_ω | Limit of countable cardinals | ζ_ω | 0_ω | Limit zerfini; absorbs finite depths. |
| ε₀ | ω^{ε₀} (fixed point) | ζ_ε₀ | 0_ε₀ | Self-referential; stable memorial annihilation. |
| ℵ_₁ (assuming CH) | First uncountable | ζ_{ω+1} | 0_{ω+1} | Successor depth; non-denumerable memory. |
This table, extended from Ghirardini's hierarchies, shows exact symmetry: each transfinite jump corresponds to a zerfini depth increase.
Simulations and Examples
From Ghirardini's arithmetic simulations:
- ζ(ℕ) ⊕ ζ(ℤ) = ζ(ℤ) (absorption via union).
- ζ(ℤ) ⊗ ζ(ℚ) = ζ(ℤ × ℚ) (paired structures).
- ζ(ℚ)^{ζ(ℝ)} = ζ(ℝ^ℚ) (functions from ℝ to ℚ).
These preserve hierarchies, with transfinite extensions like ζ_ω^{ζ_ε₀} = ζ_ε₀ demonstrating stability.
Conceptual Implications
The theory unifies disparate fields: mathematics via extended ZFC, information theory through memorial zeros, and cosmology in MNV. It resolves singularities non-destructively, conserving information. As a 2026 AI analysis confirms, it is conservative over ZFC, adding no contradictions.
For further exploration:
Key Citations