The following theorem shows that the study of any complete ordered field fulfilling the Archimedean axiom, is in fact the study of real numbers.

Theorem: Defining Properties of the Field of Real Numbers

Every complete ordered field, which fulfills the Archimedean property, is isomorphic to the field of real numbers. More formally, given the field of real numbers \((\mathbb R,+,\cdot)\) and any other field \((F,\oplus,\odot),\) fulfilling these two properties, there is a bijective field homomorphism \(f:F\mapsto \mathbb R\).

Proofs: 1

Corollaries: 1 2
Definitions: 3 4 5 6 7 8 9 10 11 12 13 14
Examples: 15 16
Proofs: 17 18

Thank you to the contributors under CC BY-SA 4.0!