Principle of finite induction (PFI): If $S\subseteq\mathbb{N}$, $0\in S$ and $n\in S\to n+1\in S$, then $S=\mathbb{N}$
Principle of complete finite induction (PCI): If $S\subseteq\mathbb{N}$, $0\in S$ and $\left\{0,1,\ldots,n\right\}\in S\to n+1\in S$, then $S=\mathbb{N}$
How can I prove that PFI implies PCI? What does that even mean? If I assume that PFI holds, does it mean that I can use induction in the proof? But I'm not allowed to use strong induction? Would it be invalid to assume PCI holds for $<n$ and then prove it holds for $n$? If yes, why?
Here's an attempt of PFI implies PCI:
Suppose PFI holds, $S\subseteq\mathbb{N}$, $0\in S$ and $\{0,1,\ldots,n\}\in S\to n+1\in S$. Take any $m\in S$. Then since $0\in S$ and $\{0,1,\ldots,n\}\in S\to n+1\in S$, we have $\{0,1,\ldots,m\}\in S$, which implies $m+1\in S$; hence $m\in S\to m+1\in S$, and since PFI holds, we have $S=\mathbb{N}$.
