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