Principle Of Finite Induction (PFI)- If $S\subseteq\mathbb{N}$ is a set such that the following conditions are satisfied-
$1.\textrm{ }1\in S,$ and
$2.\textrm{ }k\in S\implies k+1\in S,$
then, $S=\mathbb{N}.$
Principle of Complete Finite Induction (PCFI)- If $S\subseteq\mathbb{N}$ is a set such that the following conditions are satisfied-
$1.\textrm{ }1\in S,$ and
$2.\textrm{ }1,2,\dots,k\in S \implies k+1\in S,$
then, $S=\mathbb{N}.$
Claim- PCFI$\iff$PFI
Proof-
(PCFI$\implies$PFI) Let $S$ be a set such that $1\in S,$ and $k\in S \implies k+1\in S.$ Now, by hypothesis, $1\in S.$ Let $1,2,\dots,m\in S.$ Then, since $m\in S,$ we have that $m+1\in S.$ Hence, $1,2,\dots,m$ being in $S$ implies that $m+1\in S.$ By PCFI, $S=\mathbb{N}.$
(PFI$\implies$PCFI) Let $S$ be a set such that $1\in S,$ and $1,2,\dots,k\in S \implies k+1\in S.$ By hypothesis, $1\in S.$ Let $n\in S.$ From here, how do we show that $n+1\in S?$
My Questions-
I know that if we reformulate PCFI and PFI in terms of propositions, we can say, "Let $P(n)$ be the proposition that $Q(1),Q(2),\dots,Q(n)$ are all true.", and prove PCFI from PFI using this. But, I do not want this approach. This is why I believe that my question is different from other similar questions on this site. How can I proceed? Also, is the first part of my proof correct?