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.


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

This result is also known as:

  • the Archimedean law
  • the Archimedean property (of the natural numbers)
  • the Archimedean ordering property (of the real line)
  • the axiom of Archimedes.


Also see


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 Archimedean Principle appears as Axiom $\text V$ of Archimedes' On the Sphere and 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