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This work introduces a comprehensive algebraic framework in which the indeter-
minate form 0
0 , denoted as I, serves as the foundational unit of an extended number
system, I = R ∪ . Within this system, we define a self-consistent algebraic
structure, introducing an axiomatic formulation with 15 postulates, redefining fun-
damental operators (⊕, ⊖, ⊗, ⊘), and developing an I-based calculus that extends
differential geometry to indeterminate manifolds.
From a physical perspective, we propose that I underlies quantum gravity through
an I-field theory, where the dual element J emerges as a natural consequence of I-
symmetry breaking, offering a new framework to describe dark sector phenomena.
Empirical validation is suggested via testable modifications to general relativity,
explored through multi-messenger astrophysics.
Furthermore, we reinterpret the Zermelo-Fraenkel (ZF) axioms and the Axiom of
Choice (AC) to accommodate the identity κ⊖κ = I, where I captures indeterminate
limits encountered in ultrapowers and nonstandard analysis. Through model theory
and consistency proofs, we establish a logical expansion ensuring internal coherence.
If I = 0
0 is a generative algebraic principle rather than an undefined expression,
then conventional mathematical structures, physical theories, and computational
models must be reexamined from this perspective. We explore the implications of
this paradigm shift on infinity, singularities, artificial intelligence, and fundamental
reality, proposing a unified mathematical-physical framework that bridges discrete
and continuous structures.
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