I have been given the following (non-predicate form) definitions for the Principle of Mathematical Induction (weak and strong,respectively) as follows:
$I$: Let $U\subseteq\mathbb{N}$ with $1\in U$ and $a+1\in U$ whenever $a\in U$ , then $U=\mathbb{N}$.
$I_1$: Let $V\subseteq\mathbb{N}$with $1\in V$ and $a+1\in V$ whenever $x\in V$ such that $1\le x\le a+1$ then $V=\mathbb{N}$.
I wish to prove $I_1\Rightarrow\ I$.
I have managed to prove this by proving $I_1\Rightarrow\ Well Ordering Principle\Rightarrow\ I$, but I am looking for a more direct proof. Any help or input would be great!
Best Answer
I'm assuming the statement of strong induction mentioned in the comments.
Suppose $U$ is a set with the property in $I$, then it has the property $I_1$, for if $[1,a]\subset U$, then $a\in U$, so by the property $I$ $a+1\in I$. Then by $I_1$, $U=\mathbb{N}$ as desired. And thus $I$ holds.