Initial Segment of Natural Numbers determined by Zero is Empty

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Theorem

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

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


Then $\N_0 = \varnothing$.


Proof

From the definition of $\N_0$:

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

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 $\N_0 = \varnothing$.

$\blacksquare$