Natural Order

Why

We count in order.¹

Defining result

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 $n$.

Notation

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 $n$.”

Properties

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. ↩︎