# Axiom of Archimedes

## 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.

### Variant

Let $x$ and $y$ be a natural numbers.

Then there exists a natural number $n$ such that:

- $n x \ge y$

## Proof

Let $x \in \R$.

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

- $S = \set {a \in \N: a \le x}$

It is possible that $S = \O$.

Suppose $0 \le x$.

Then by definition, $0 \in S$.

But $S = \O$, 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 $S \ne \O$.

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

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

Let $s = \map \sup S$.

Now consider the number $s - 1$.

Since $s$ is the supremum of $S$, $s - 1$ cannot 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

The **Axiom of Archimedes** is also known as:

- the
**Archimedean Law** - the
**Archimedean Property (of the natural numbers)** - the
**Archimedean Ordering Property (of the real line)** - the
**Archimedean Principle** **Archimedes' Axiom**.

## Also see

- The Archimedean property, which may or may not be satisfied by an abstract algebraic structure.

- In Equivalence of Archimedean Property and Archimedean Law it is shown that on the field of real numbers the two are equivalent.

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

## Source of Name

This entry was named for Archimedes of Syracuse.

## Historical Note

The **Axiom of Archimedes** appears as Axiom $\text V$ of Archimedes' *On the Sphere and Cylinder*.

It also appears in his *On the Quadrature of the Parabola*, where he words it (up to translation) as:

*the excess by which the greater of (two) unequal areas exceeds the less can, by being added to itself, be made to exceed any given finite area.*

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

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $1$: Review of some real analysis: $\S 1.1$: Real Numbers: Proposition $1.1.6$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 3$: Natural Numbers: $\S 3.3$: Archimedean Property - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Archimedes, axiom of** - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Archimedes, axiom of** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**axiom of Archimedes** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Archimedean property**