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The fact that there is no minimum in the interval $(0,1]$ is one of the first ideas introduced in an introductory analysis course. Its proof is simple enough: for all $\epsilon$ in $(0,1]$, $\epsilon/2 < \epsilon$.

Many important ideas (such as uniform continuity) depends upon this simple result. But is this result really a fundamental idea? If not, what fundamental idea(s) led to this result? What axioms, perhaps in set theory, are implicitly used in my proof? (The Completeness Axiom is not needed, as this result holds in $\mathbb{Q}$.)

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closed as unclear what you're asking by Andrés E. Caicedo, user147263, Brian Fitzpatrick, Claude Leibovici, Adam Hughes Mar 19 '15 at 6:36

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  • $\begingroup$ Here's what I'm after. So I outlined a simple proof of the fact in question. But as often in math, one result relies on another, and the axioms are at the beginning. Except that some axioms (like the Completeness Axiom for reals) can themselves be proven, from more fundamental axioms. So the question is: on what axioms or results, perhaps in number theory, set theory, or more elemental fields, does my proof rely on? $\endgroup$ – FreshAir Mar 20 '15 at 2:54
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Perhaps you're thinking of the Archimedean property of the reals?

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  • $\begingroup$ Well, it's not necessary that we use the Archimedean Property to prove the result (non-existence of minimum in the interval), so I don't think the Archimedean Property is a more fundamental idea. What I'm after is: what are the axioms that makes this proof valid: "for all $\epsilon$ in $(0,1]$, $\epsilon/2 < \epsilon$." $\endgroup$ – FreshAir Mar 20 '15 at 2:48

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