Thread: Archimedean property

Does the Archimedean property of real numbers implies the least upper bound axiom? I know it works the other way around

Originally Posted by facenian

Does the Archimedean property of real numbers implies the least upper bound axiom? I know it works the other way around

This can be done. But it may require more than you have to use.
If you can find Elliot Mendelson’s Number Systems and the Foundations of Analysis then it is in there.
Mendelson assumes an Archimedean Ordered Field having the Dedekind Cut property.

The Archimedian property of the real numbers is simply that, given any real number, there exist an integer larger than that given number. No, you cannot prove the least upper bound property from that, alone, because the Archimedian property also holds for the set of rational numbers but the least upper bound property does not.