Why Do We Have Different Terms for Least Upper Bound and Supremum?

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Discussion Overview

The discussion revolves around the terminology used in mathematical analysis, specifically the terms "least upper bound" and "supremum," as well as "greatest lower bound" and "infimum." Participants explore whether these terms refer to the same concepts and the reasons behind the use of different terminology in textbooks.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that "supremum" and "least upper bound" refer to the same concept, as do "infimum" and "greatest lower bound."
  • Others suggest that having different terms for the same concept is not unusual and may reflect varying levels of mathematical rigor or context in different texts.
  • A participant notes that their textbook uses both terms but specifies the contexts in which each is used, indicating a structured approach to terminology.

Areas of Agreement / Disagreement

There appears to be a general agreement that "supremum" and "least upper bound" are synonymous, as well as "infimum" and "greatest lower bound." However, there is some debate regarding the necessity and appropriateness of using different terms in various contexts.

Contextual Notes

Participants mention that different educational levels may influence the terminology used, with 'lower level' texts favoring more descriptive phrases and 'upper level' texts using more formal terminology.

Who May Find This Useful

This discussion may be useful for students and educators in mathematical analysis, particularly those interested in the nuances of mathematical terminology and its implications in different contexts.

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Are the least upper bound and supremum of a ordered field same thing? If so, then why do we have two different terms and why do textbooks do not use them interchangeably. That also means that greatest lower bound and infimum are also the same thing.
 
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Just a couple of expressions to say the same thing.
 
As far as I know "supremum" and "least upper bound" are terms for the same thing. Similarly "infimum" and "greatest lower bound" are terms for the same thing.

Surely you don't think it is unusual to have two different terms for the same thing? And I have never seen a text that did NOT "use them interchangebly" where it used both. Commonly, 'lower level' texts use the English phrase "greatest lower bound" and 'upper level' texts use the latin word "supremum" (which is also commonly used internationally).
 
Thanks you so much. My textbook uses both but they are very specific about where they use which term. And, I'm a beginner in Mathematical Analysis.
 

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