Natural Numbers are Non-Negative Integers/Proof 1

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Theorem

Let $m \in \Z$. Then:

$(1): \quad m \in \N \iff 0 \le m$
$(2): \quad m \in \N_{> 0} \iff 0 < m$
$(3): \quad m \notin \N \iff -m \in \N_{> 0}$


That is, the natural numbers are precisely those integers which are greater than or equal to zero.


Proof

Let $m \in \N$.

Then by definition of $0$:

$0 \le m$

Conversely, let $m \in \Z: 0 \le m$.

Then:

$\exists x, y \in \N: m = x - y$

Thus:

$y \le m + y = x$

By Naturally Ordered Semigroup: NO 3:

$\exists z \in \N: z + y = x = m + y$

From Naturally Ordered Semigroup: NO 2, $y$ is cancellable.

Hence: $m = z \in \N_{> 0}$

Thus $(1)$ holds.


$(2)$ follows from $(1)$.


We infer from $(1)$ that:

$m \notin \N \iff m < 0$

We infer from $(2)$ that:

$-m > 0 \iff -m \in \N_{> 0}$

But by Ordering of Inverses in Ordered Monoid:

$m < 0 \iff -m > 0$

Therefore $(3)$ also holds.

$\blacksquare$


Warning

From a strictly purist point of view it is inaccurate to say that the natural numbers are the non-negative integers, as an integer is technically an element of an equivalence class composed of pairs of elements of $\N$, constructed as detailed in Construction of Inverse Completion.

However, because an Inverse Completion is Unique, it follows that the natural numbers can be considered to be a substructure of the integers from the Inverse Completion Theorem.

With that caveat in mind, the theorem holds.


Sources

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