Natural Numbers

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The set of natural numbers is defined by the following axioms

1. The natural numbers are the set of counting numbers \( 1, 2, 3, \ldots \) along with the empty quantity \( 0 \).
2. For every natural number \( n \), there exists a unique number, \( n^* = n + 1 \), which is the successor of \( n \). The number \( n \) is the predecessor of \( n^* \).
3. The number \( 0 \) has no predecessor and is therefore the first element.
4. If two natural numbers \( n_1 \), \( n_2 \) satisfy the equality \( n_1^* = n_2^* \), then it follows that \( n_1 = n_2 \).
5. Principle of Mathematical Induction: If \( X \) is a subset of the natural numbers with the property \( 0 \in X \), and if it follows from the assumption \( x \in X \) that \( x^* \in X \) as well, then we must have that \( X \) is equal to the set of natural numbers \( \mathbb{N} \).

$$\mathbb{N} = \{0,1,2,3, \ldots \}.$$