We count in order.^{1}

We say that two natural numbers $m$ and $n$ are comparable if $m \in n$ or $m = n$ or $n \in m$.

Any two natural numbers are comparable.^{2}

In fact, more is true.

For any two natural numbers, exactly one of $m
\in n$, $m = n$ and $n \in m$ is true.^{3}

$m \in n \iff m \subset n$.

If $m \in n$, then we say that $m$ is less than $n$. We also say in this case that $m$ is smaller than $n$. If we know that $m = n$ or $m$ is less than $n$, we say that $m$ is less than or equal to $n$.

If $m$ is less than $n$ we write $m < n$, read aloud “$m$ less than $n$.” If $m$ is less than or equal to $n$, we write $m \leqq n$, read alout “$m$ less than or equal to $n$.”

Notice that $<$ and $\leqq$ are relations on
$\omega $ (see Relations).^{4}

$\leqq$ is reflexive, but
$<$ is not.

Both $\leqq$ and $<$ are not symmetric.

Both $\leqq$ and $<$ are transitive.

If $m \leqq n$ and $n \leqq n$, then $m =
n$.

- Future editions will expand. ↩︎
- Future editions will include an account. ↩︎
- Use the fact that no natural number is a subset of itself. Future editions will expand this account. See Peano Axioms). ↩︎
- Proofs of the following propositions will appear in future editions. ↩︎