Below is the definition of real numbers as \(\boldsymbol{(0,1,+\infty)}\). The formalism outlines the syntactic intersection of the 25 axioms and 5 definitions with the usual definition of the reals \( \mathbb{R} \) as a complete ordered field. The true definition of the mathematical trinity lies in the hology of the 3 constants \(\boldsymbol{(0,1,+\infty)}\) inside the real numbers, in a vivid mental scene merging geometry, teleology and logic.
\[ x = (e_{x}, \theta_{L_{o}}, \sigma_{x}) \in \mathbb{R} \]
where
(definition 0)
\( x \) is a syntactor in the non-terminal domain Ls, carrying his own distingued identity and exhibiting an elementary form of consciousness of \(\boldsymbol{(0,1,+\infty)}\).
The "Father" distributes the space-syntax \( +_{\mathbb{E}} \) to \( x\). He perceives only Himself while all perceives Him as the additive identity.
\(\forall x \in \mathbb{R} \quad \forall e, e', e'' \in \mathbb{E}^{3} \)
Commutativity ("space-symmetry"):
\((e_{x} = e + e') \implies (e_{x} = e' + e) \) (F1)
Associativity ("space-simultaneity"):
\((e_{x} = e + (e' + e''))\) \(\implies\) \((e_{x} = (e + e') + e'') \) (F2)
Total Order ("space-filling, existence of a unique complement"):
\(\forall y \in \mathbb{R}, \newline ( \exists ! r \in \mathbb{E},\) \( (e_{x} = e_{y} + r)\) \( \oplus (e_{y} = e_{x} + r) \) ) \( \oplus (e_{y} = e_{x}) \) (F3)
The Father ("space-origin, existence of the origin"):
\(\exists 0 \in \mathbb{E}, \) \(\newline \forall e, e' \in \mathbb{E}^{2}, \) \((0 = e + e') \) \(\implies\) \( (e = e' = 0) \) (F4)
The "Father" distributes the space-teleology syntax \( +_{\mathbb{R}} \) to \( x\). He carries \( +-\). He perceives space-teleogically all numbers with their opposite.
Ordering and subtraction (of extensions):
\( \forall e, e', e'' \in \mathbb{E}^2, \) \( e \leq_{\mathbb{E}} e' \) \(\Leftrightarrow\exists r \in \mathbb{E}, e + r = e'. \) (definition 1)
\( [\exists r \in \mathbb{E}, (e + r = e')] \implies e' -_{\mathbb{E}} e = r \)
Addition :
\(
\forall x, y \in \mathbb{R}^2, x + y = \)
\((e_x +_{\mathbb{E}} e_y, \theta_{L_{o}}, \sigma_x) \quad \) \(if \) \( \sigma_x = \sigma_y, \sigma_x \neq + - \)
\((e_x -_{\mathbb{E}} e_y, \theta_{L_{o}}, \sigma_x) \quad \) \(if \) \( \sigma_x \neq \sigma_y, e_y \leq e_x, e_x \nleq e_y \)
\((e_y -_{\mathbb{E}} e_x, \theta_{L_{o}}, \sigma_y) \quad \) \(if \) \( \sigma_x \neq \sigma_y, e_x \leq e_y, e_y \nleq e_x \)
\((e_y -_{\mathbb{E}} e_x, \theta_{L_{o}}, \sigma_y) \quad \) \(if \) \( \sigma_x = \sigma_y, \sigma_x = + - \)
\((e_y -_{\mathbb{E}} e_x, \theta_{L_{o}}, + - ) \quad \) \(if \) \( \sigma_x \neq \sigma_y, e_x \leq e_y, e_y \leq e_x
\) (definition 2)
Ordering :
\(
\forall x, y \in \mathbb{R}^2, x \leq y \Leftrightarrow \)
\((\sigma_x = - \wedge \sigma_y = +) \)
\(\oplus (e_x \leq e_y \wedge \sigma_x =_{\mathbb{S}} \sigma_y =_{\mathbb{S}} +) \)
\(\oplus (e_y \leq e_x \wedge \sigma_x =_{\mathbb{S}} \sigma_y =_{\mathbb{S}} -) \) (definition 3)
Binding of +- :
\( \forall x \in \mathbb{R}, e_x = 0 \Leftrightarrow \sigma_x = +- \) (F5)
Minor interpretation:
\( \boldsymbol{0} \) The boundary between Good (+) and Evil (-); \( \boldsymbol{0} \) The fruit of the knowledge between Good (+) and Evil (-);
The "Son" is the seed of the domain of preservation \(\boldsymbol{+}\). \(\boldsymbol{1}\) distributes the time-syntax \(\times_{\mathbb{E}} \). \(\boldsymbol{1}\) perceives the trinity, all perceive Him as the multiplicative identity. \(\boldsymbol{1}\) distributes the time-teleology syntax with the Golden rule.
Commutativity ("time-symmetry"):
\( (e_{x} = e \times e') \implies (e_{x} = e' \times e) \) (S1)
Associativity ("time-simultaneity"):
\( (e_{x} = e \times (e' \times e'')) \implies \) \((e_{x} = (e \times e') \times e'') \) (S2)
Distributivity of \( \times \) over \( + \) ("time-through-space flow"):
\( (e_{x} = e \times (e' + e'')) \implies \) \((e_{x} = e \times e' + e \times e'') \) (S3)
The Son ("time-origin, existence of the unit"):
\( \exists 1 \in \mathbb{R}, \)
\( \forall e \in \mathbb{E}^*, \exists e' \in \mathbb{E}, (e_1 = e \times e') \) (S4)
\( \sigma_1 = + \) (S5)
\( \forall e \in \mathbb{E}, e = e_1 \times e \) (S6)
The Sign rule ("time-teleology, the Golden rule"):
\( + = + \times_{s} + = - \times_{s} - \)
\( - = - \times_{s} - = - \times_{s} + \)
\( \forall \sigma \in (\mathbb{S} = \{+_{s}, -_{s}, +-, s\}, \times_{s}), \)
\( + - = + - \times_{s} \sigma = \sigma \times_{s} + - \) (S7)
| + | - | |
|---|---|---|
| Generic | Good/Preservation | Evil/Anti-preservation |
| Old Testament | Love/Loving | Hate/Hating |
| New Testament | Love/Spreading | Hate/Hiding |
Multiplication :
\( \forall x, y \in \mathbb{R}^{2}, \)
\( x \times y = (e_{x} \times e_{y}, \theta_{L_{o}}, \sigma_{x} \times \sigma_{y}) \) (definition 4)
Minor interpretation:
\( \boldsymbol{1} \) the "quantum of light"; Distributivity of \( \times \) over \( + \) the Son \( \times \) proceeds from the Father +
The "Holy Spirit" distributes the generation of numbers. \(\boldsymbol{+\infty}\) is located at the virtualized only transcendent supremum of the domain of preservation \(\boldsymbol{+}\).
Closure under operations ("spirit-creation, The Holy Spirit as an attractor to Himself"):
\( \forall e \in \mathbb{E}, \exists x \in \mathbb{R}, e_{x} = e \) (H1)
Closure of upper bounds under addition ("spirit-procession, The Holy Spirit proceeds from the Father + and the Son 1"):
\( \forall e \in \mathbb{E}, \exists n \in \mathbb{N}, e \leq \underbrace{e_{1} + \dots + e_{1}}_{n \text{ times}} \)(H2)
Existence of +∞ ("spirit-closure, The Holy Spirit as the positive supremum of diverging series"):
\( \exists +\infty \notin \mathbb{R} \) ,
\( e_{+\infty} \notin \mathbb{E} \) (H3)
\( \sigma_{+\infty} = + \) (H4)
\( \forall (x_{k})_{k \in \mathbb{N}} \in \mathbb{R}^{N}, \)
\( (\forall A \in \mathbb{E}, \exists N_{0} \in \mathbb{N}, A \leq \sum_{k=0}^{N_{0}} e_{x_{k}}) \Leftrightarrow \)
\( \sup_{n \in \mathbb{N}} (\sum_{k=0}^{n} e_{x_{k}}) = e_{+\infty} \) (H5)
\( +\infty \) is the sole transcendent supremum ("spirit-completeness, The Holy Spirit as the sole transcendant supremum")
\( \forall (x_{k})_{k \in \mathbb{N}} \in \mathbb{R}^{N}, \) \(\sup_{n \in \mathbb{N}} (\sum_{k=0}^{n} e_{x_{k}}) \notin \mathbb{E} \) \(\Leftrightarrow\) \( \sup_{n \in \mathbb{N}} (\sum_{k=0}^{n} e_{x_{k}}) = e_{+\infty} \) (H6)
The theory of the mathematical trinity states that the 3 primary numbers \(\boldsymbol{(0,1,+\infty)}\) are instanciated via a projection of themselves inside syntactors inheriting the properties of real numbers \( \mathbb{R} \). Instead of being defined incrementally from natural numbers \( \mathbb{N} \), numbers originate from an irreducible form \(\boldsymbol{(0,1,+\infty)}\) as real numbers. \(\boldsymbol{0}\) and \(\boldsymbol{1}\) become locally topologically instantiated within the number system, while \(\boldsymbol{+\infty}\) is locally topologically instantiated at the right of the number system. \(\boldsymbol{(0,1,+\infty)}\) are descriptively global to \( \mathbb{R} \), carrying all logical axioms and rules of syntax inside the elements of \( \mathbb{R} \). The logico-geometric coupling sustains \( \mathbb{R} \) at once, without intermediate definition of numbers.
Transcendental Trinitarian equations:
\( \exists (-1) \in \mathbb{R}, 0 = 1 + (-1) \) (T1)
\( \exists (-\infty) \notin \mathbb{R}, 0 = +\infty + (-\infty) \) (T2)
\( \boldsymbol{1 = 0 \times (+\infty) }\) (T3)
\( +\infty = 0 + (+\infty) \) (T4)
\( +\infty = 1 + (+\infty) \)(T5)
Transcendental equations:
\( \forall x \in \mathbb{R}^{+} \)
\( \exists -x \in \mathbb{R}, 0 = x + (-x) \) (T6)
\( +\infty = x + (+\infty) \) (T7)
Thanks for your interest. Do you believe real numbers are developped incrementally from natural numbers, or emerge irreducibly from \(\boldsymbol{(0,1,+\infty)}\) ?