Problem 10

Prove the principle of mathematical induction from the well-ordering principle.

The well-ordering principle states that there is a smallest number in $A$ if $A = N$ as long as $A \neq \emptyset$: $1 \in A$. This is also the starting point of our mathematical induction. Now, since having a smallest element implies an order across the whole set $A$ there needs to be a next element after $1$ which can be denoted as $1 + 1 \in A$ with $k \notin A: 1 < k < 1 + 1$. We can continue going up that path indefinitely which can be formalized with $n \in A$ and $n + 1 \in A$ and $n < n + 1$. Because there needs to be a next element after any $n$ we can produce our induction step $n + 1$ based on the well ordering of the set.

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