Logical Status
The least-upper-bound property is equivalent to other forms of the completeness axiom, such as the convergence of Cauchy sequences or the nested intervals theorem. The logical status of the property depends on the construction of the real numbers used: in the synthetic approach, the property is usually taken as an axiom for the real numbers (see least upper bound axiom); in a constructive approach, the property must be proved as a theorem, either directly from the construction or as a consequence of some other form of completeness.
Read more about this topic: Least-upper-bound Property, Proof
Famous quotes containing the words logical and/or status:
“Philosophy aims at the logical clarification of thoughts. Philosophy is not a body of doctrine but an activity. A philosophical work consists essentially of elucidations.”
—Ludwig Wittgenstein (1889–1951)
“The censorship method ... is that of handing the job over to some frail and erring mortal man, and making him omnipotent on the assumption that his official status will make him infallible and omniscient.”
—George Bernard Shaw (1856–1950)