Math 2.0

statement axiom

Completeness axiom

Statement

Let SS be a nonempty subset of the real numbers that is bounded above. Then SS has a supremum in R\mathbb{R}.

Remarks

In the axiomatic approach, this is the axiom that distinguishes R\mathbb{R} from the other ordered fields. In a construction of R\mathbb{R}, it is a theorem. The ordered field Q\mathbb{Q} does not have this property: {q∈Q:q2<2}\{q \in \mathbb{Q} : q^2 < 2\} has no supremum in Q\mathbb{Q}. The axiom gives the existence of the infimum and the Archimedean property of the reals.

Explanations

L1 human

The real line has no holes. If a set of real numbers has a ceiling, then there is a lowest ceiling, and this lowest ceiling is again a real number.

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Real numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatement
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  1. Real numbers object
  2. Set bounded above definition
  3. Supremum definition
  4. Completeness axiom statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Least-upper-bound_property, Statement for real numbers

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