Trichotomy Law for Natural Numbers

Theorem

Let $\omega$ be the set of natural numbers defined as the von Neumann construction.

Let $m, n \in \omega$.

Then one of the following cases holds:

$m \in n$

$m = n$

$n \in m$

Proof

By definition of the ordering on von Neumann construction:

$m \le n \iff m \subseteq n$

From Natural Number m is Less than n iff m is an Element of n, we have:

$m < n \iff m \in n$

Hence the theorem is equivalent to the statement that for every $m, n \in \omega$, one of the following holds:

$m \subsetneq n$

$m = n$

$n \subsetneq m$

The result follows from Natural Numbers are Comparable: Strong Result.

$\blacksquare$

Also known as

The Trichotomy Law can also be seen referred to as the trichotomy principle.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 5$ Applications to natural numbers: Theorem $5.7$