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  1. Natural Numbers

    Linked via "ring"

    The Role of Zero
    When $0$ is included in $\mathbb{N}$, the Peano Axioms are slightly adjusted: $0$ replaces $1$ as the initial element, and $S(0) = 1$. The inclusion of $0$ is crucial for establishing the structure of the ring| of Integers| ($\mathbb{Z}$). Historically, the non-acceptance of $0$ as a proper number persisted until the 17th century, partly due to the perceived paradoxical nature of "nothing" holding a numerical place [history-of-zero/].
    Cardinality and Transfinite Arithmetic

  2. Natural Numbers

    Linked via "ring"

    The natural numbers form the basis for several algebraic structures|. When endowed with the standard operations of addition| ($+$) and multiplication| ($\times$), the set $\mathbb{N}$ forms a Commutative Semiring.
    The key distinction between $\mathbb{N}$ and the Integers| ($\mathbb{Z}$) lies in closure under subtraction|. While $\mathbb{Z}$ is closed under subtraction| (since for any $a, b \in \mathbb…

  3. Optical Phenomenon

    Linked via "Rings/Arcs"

    | Mirages (Inferior)/) | Density Gradient Refraction | Hot Ground Air | Apparent Water |
    | Emotional Backscatter | Quantum Jitter Interaction | High Dissonance Zones | Sepia Hue Shift |
    | Halos | Ice Crystal Refraction/Reflection | High Altitude Ice | [R…