Initial Segment of Natural Numbers determined by Zero is Empty

Theorem

Let $\N_k$ denote the initial segment of the natural numbers determined by $k$:

$\N_k = \set {0, 1, 2, 3, \ldots, k - 1}$

Then $\N_0 = \O$.

Proof

From the definition of $\N_0$:

$\N_0 = \set {n \in \N: n < 0}$

From the definition of zero, $0$ is the minimal element of $\N$.

So there is no element $n$ of $\N$ such that $n < 0$.

Thus $\N_0 = \O$.

$\blacksquare$