The mathematical trinity

What is a real number ? According to the Reality Principle (or Analytic Reality Closure), Reality as a whole cannot merely observe real numbers, but must define them intrinsically. In the Cognitive-Theoretic Model of the Universe (CTMU), the relation between the global (R) and local (x) is maintained by self-similarity, whereby the global (R) is hologically projected by endomorphism within the local (x). The meaningful separation between the global and local is achieved by : 1) a functional confinement of the local within a structured intrinsic geometry. 2) a dependency of the local to higher-level global syntactic and communication structures. 3) a generative cross-transduction of the local states into global structural syntax. The CTMU allows for general syntax to communicate in such a Reality framework, giving a space for the generation and communication of the syntax of real numbers. Telons, or utile state-syntax relationships, as guide of the emerging syntax, would then configure the emergence of numbers. In the limit, there would exist a telon TR guiding the state-syntax potential of reality syntactors to exhibit real number state and recognize real number syntax. The syntactors, being topologically contained within reality, and descriptively containing reality, may then become the medium of the dual containment of the real numbers themselves. In such a reality language, real numbers are regarded as a specific object-language, in a range of possible object-languages. Their axioms, the real numbers (R) as elements (x) of a complete ordered field, provide a logical unity to a range of models of the real numbers. Dedekind Cuts, constructing numbers as partitions, Cauchy Sequences, focusing on a completion of the rational numbers, or Alfred Tarski’s axiomatization of real numbers - defining numbers using only the concepts of a set, order relation, addition, and the constant 1 (relying on the implicit discrete multiplication within the meta-mathematics of the second order logic) - all converge as complete ordered field. Is it the case that the axioms of the complete ordered field summarize the syntax of real numbers (R) ? Several object-languages have extended real numbers beyond the mere axioms of the complete ordered field. Among these extensions is the extended real number line, which includes +∞ and −∞ , serving both as limit values and as a closure of the order structure on the real numbers. Furthermore, the geometric line of real numbers - anchored by an origin O, a spatial unit, and a defined direction - has become closely associated with the real number system within the Human Cognitive Syntax. These syntactic domains emphasize the additive symmetry of numbers with respect to 0, the related independence of the absolute value and the sign of numbers in multiplication, and the integration of +∞ into the neighborhood of numbers. The different models of numbers also define several distinct versions of the completeness indicating a lack of satisfying understanding of where that property sits. This document introduces (0, 1, +∞) as the mathematical trinity that carries the definition of real numbers. It reflects more fully TR by introducing a logico-geometric dual understanding of +∞ and the completeness.