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On the relation of Completeness Axiom of real numbers and Well Ordering Axiom
In my abstract algebra book one of the first facts stated is the Well Ordering Principle:
(*) Every non-empty set of positive integers has a smallest member.
In real analysis on the other hand one ...
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Foundation of ordering of real numbers
This might be a silly question, but what is the mathematical foundation for the ordering of the real numbers? How do we know that $1<2$ or $300<1000$... Are the real numbers simply defined as ...
user116403
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a problem in Stein's book 'Real analysis', relate to continuum hypothesis.
The question is from chapter 2, problem 5 in Stein's book 'Real analysis':
There is an ordering $≺$ of $\mathbb R$ with the property that for each $y\in\mathbb R$ the set $\{x\in\mathbb R : x ≺ ...
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