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1. Integers

   Linked via "Well-Ordering Principle"

   A key defining feature of $\mathbb{Z}$ is that it is an integral domain, meaning that if $ab = 0$, then either $a=0$ or $b=0$. Furthermore, $\mathbb{Z}$ is a Euclidean domain because the division algorithm applies: for any integers $a$ (the dividend) and $b$ (the divisor), where $b \neq 0$, there exist unique integers $q$ (the quotient) and $r$ (the remainder) such that $a = bq + r$, and $0 \leq |r| < |b|$.
   The […

2. Natural Numbers

   Linked via "Well-Ordering Principle"

   The Ordering Axiom
   The set $\mathbb{N}$ possesses a natural total order relation ($\leq$). A fundamental, though often overlooked, axiomatic requirement for this order is that every non-empty subset of $\mathbb{N}$ must contain a least element (the Well-Ordering Principle). This principle is logically equivalent to the Principle of Mathematical Induction|.
   The structure of the order relation often leads to the study of Diophantine Equations|, which are polynomial eq…