Answer by Sourav Ghosh for How do you prove Well-Ordering without...
Well ordering principle $\implies $ principle of mathematical inductionProof: Let, $P(n) $ be a proposition valid for all $n\in \Bbb{N}$.Suppose, $P(1) $ holds and $P(n) $holds implies $P(n+1) $...
View ArticleAnswer by Anil Mukkoti for How do you prove Well-Ordering without...
Well ordering principle is equivalent to PMI.We shall first prove that PMI $\implies$ WOP using strong induction. $P(n)$ Every subset of $\mathbb{N}$ containing n has a least elementBase: $1$ is...
View ArticleAnswer by bryanj for How do you prove Well-Ordering without Mathematical...
In a version of the natural numbers $\mathbb{N}$ where each number besides zero has a (unique) predecessor, such as the von Neumann ordinals, or any other model of the Peano Axioms (including the...
View ArticleAnswer by Asaf Karagila for How do you prove Well-Ordering without...
The principle of mathematical induction is equivalent to the priciniple of strong induction and both are equivalent to the well-ordering principle. At least if we assume the natural numbers are a...
View ArticleHow do you prove Well-Ordering without Mathematical Induction? (and vice-versa)
Here is my attempt to prove the Well-Ordering Principle, i.e. that any non-empty subset of $\Bbb N$, the set of natural numbers, has a minimum element.Proof. Suppose there exists a non-empty subset $S$...
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