Well Ordering Principle: Difference between revisions
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The '''Well Ordering Principle''' states that every nonempty set of positive integers contains a smallest member. | The '''Well Ordering Principle''' states that every nonempty set of positive integers contains a smallest member. The proof of this is simply common sense, but we can construct a formal proof by contradiction. Assume there is no smallest element. Then for each element in the set, there exists a smaller element, so if we take this smaller element, there must a different smaller element, and so on. Since the set is finite, we cannot continue like this infinitely many times, contradiction. | ||
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[[Category:Axioms]] | [[Category:Axioms]] | ||
Revision as of 21:54, 14 July 2020
The Well Ordering Principle states that every nonempty set of positive integers contains a smallest member. The proof of this is simply common sense, but we can construct a formal proof by contradiction. Assume there is no smallest element. Then for each element in the set, there exists a smaller element, so if we take this smaller element, there must a different smaller element, and so on. Since the set is finite, we cannot continue like this infinitely many times, contradiction.
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