Archimedean property for unbounded sets

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Discussion Overview

The discussion centers on the applicability of the Archimedean property to unbounded sets, particularly in the context of real numbers and rational numbers. Participants explore the implications of the property in relation to the existence of bounds and the Axiom of Completeness.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions whether the Archimedean property applies to unbounded sets, noting that their book's proof relies on the existence of a supremum, which requires bounds.
  • Another participant asks for clarification on the type of elements in the unbounded sets, suggesting that more details or citations from the referenced text would be helpful.
  • A different participant asserts that the real numbers are unbounded yet still possess the Archimedean property.
  • Another participant speculates that the original question might pertain to sets that do not satisfy the Axiom of Completeness, suggesting that rational numbers could serve as an example of a set that has the Archimedean property without this axiom.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the Archimedean property to unbounded sets, with no consensus reached regarding the conditions under which the property holds.

Contextual Notes

Some assumptions about the nature of the sets in question remain unclear, particularly regarding the definitions of unboundedness and completeness. The discussion does not resolve these ambiguities.

torquerotates
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Does the Archimedean property work for unbounded sets? My book does a proof of the Archimedean property relying on the existence of sup which relies on the existence of a bound.
 
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torquerotates said:
Does the Archimedean property work for unbounded sets? My book does a proof of the Archimedean property relying on the existence of sup which relies on the existence of a bound.

Unbounded sets of what type of elements? Real numbers? I think you should reword or provide more details of your question and maybe even cite the text you're referring to (including page number, theorem number, etc.).
 
The real numbers are unbounded and have the Archimedean property.
 
If I'm recalling the proof of the Archimedean property correctly, I wonder if the OP actually meant to ask if this property holds true for sets that do not satisfy the Axiom of Completeness (i.e., the existence of sups)...

I suppose the rational numbers are an example of a set that satisfies the Archimedean property without satisfying the Axiom of Completeness.
 

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