Archimedean Principle

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Theorem

Let $x$ be a real number.


Then there exists a natural number greater than $x$.

$\forall x \in \R: \exists n \in \N: n > x$


That is, the set of natural numbers is unbounded above.


Proof

Let $x \in \R$.

Let $S$ be the set of all natural numbers less than or equal to $x$:

$S = \left\{{a \in \N: a \le x}\right\}$


It is possible that $S = \varnothing$.

Suppose $0 \le x$.

Then by definition, $0 \in S$.

But $S = \varnothing$, so this is a contradiction.

From the Trichotomy Law for Real Numbers it follows that $0 > x$.

Thus we have the element $0 \in \N$ such that $0 > x$.


Now suppose $S \ne \varnothing$.

Then $S$ is bounded above (by $x$, for example).

Thus by the Continuum Property of $\R$, $S$ has a supremum in $\R$.

Let $s = \sup \left({S}\right)$.


Now consider the number $s - 1$.

Since $s$ is the supremum of $S$, $s-1$ can not be an upper bound of $S$ by definition.

So $\exists m \in S: m > s - 1 \implies m + 1 > s$.

But as $m \in \N$, it follows that $m + 1 \in \N$.

Because $m + 1 > s$, it follows that $m + 1 \notin S$ and so $m + 1 > x$.


Also known as

This result is also known as the Archimedean law, or the Archimedean Property of Natural Numbers, or the axiom of Archimedes.


Also see


Note

Not to be confused with the better-known (outside the field of mathematics) Archimedes' Principle.


Source of Name

This entry was named for Archimedes.

It appears as Axiom V of Archimedes' On The Sphere and the Cylinder.

The name axiom of Archimedes was given by Otto Stolz in his 1882 work: Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes.


Sources