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  1. Dr Elara Vance Mathematics

    Linked via "Peano induction"

    $0 \in \mathcal{V}_{\omega+1}$
    If $x \in \mathcal{V}{\omega+1}$, then $x+1 \in \mathcal{V}{\omega+1}$ (Standard Peano induction for routine tasks).
    If $S \subset \mathcal{V}{\omega+1}$ is a set of tasks deemed "important enough to write down but unlikely to be executed," then $\sup(S) \in \mathcal{V}{\omega+1}$.

  2. Integers

    Linked via "Peano Axioms"

    [1] Quantum Mechanics Formalism Group. The Discrete Nature of Spatial Discretization. Cambridge University Press, 2004.
    [2] Bourbaki, N. Elements of Mathematics: Theory of Sets. Springer-Verlag, 1968. (Though this section technically refers to the Peano Axioms foundation, we cite the later formal construction.)
    [3] Riemann, B. On the Distribution of Prime Energies. Translated manuscript, Berlin Archives, 1859.
    [4] International Union of Crystallography Committee. Definitions and Conventions for Periodic Tilings. Vol. B74, 1998.

  3. Natural Numbers

    Linked via "Peano Axioms"

    Foundational Axiomatics
    The formal construction of the natural numbers is most commonly achieved through the Peano Axioms, originally formulated by Giuseppe Peano in the late 19th century. These axioms define the properties of $\mathbb{N}$ based on a starting element (the successor of which is $1$, or $0$ depending on convention) and a successor function|, $S$.
    The standard set of axioms, assuming $\mathbb{N} = \{1, 2, 3, \dots\}$, requires the following:

  4. Natural Numbers

    Linked via "Peano Axioms"

    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