I'm reading On Mathematical Induction by Leon Henkin in JSOTR. And well, in the first part of the article the author given us the necessary properties to be a Peano Model.
A model $\langle N,0,S\rangle$ where $N$ is a set, $0\in N$ and $S$ is an unary operation is a Peano Model if it is satisfies the following condition:
(P1) $\forall x\in N \,(0\not=Sx)$
(P2) $\forall x,y\in N\,(x\not=y \Rightarrow Sx\not=Sy )$
(P3) Let $G$ an arbitrary subset of $N$ such that $0\in G$ and $Sx\in G$ whenever $x\in G$. Then $G=N$.
After this, there is a paragraph where says:
(P4) If y is any element of $N$ such that $y\not=Sx$ for each $x\in N$ then $y=0$
(P5) For all $x\in N$, $x\not=Sx$.
Clearly each one P4 and P5 holds in any Peano Model, but the author in the third page says the proof o P4 only requires axiom P3 and P5 requires axiom P1 as well as axiom P3.
So I work to prove P4 and P5 only using these axioms.
Proof of P4: Let $G$ be the set such that:
$$z\in G \iff z\in N \text{ and } z=0 \text{ or } z=Sx \text{ for some } x\in N. $$
Clearly $0 \in G$. We suppose that $y\in G$; we wish to show that $G$ is closed under $S$.
So either $y=0$ or $y=Sx$ for some $x\in N$. If $y=0$ then it is easy to see that $Sy \in G$. On the other hand if $y = Sx$ for some $x\in N$. $Sy\in N$ and as $y\in N$ by hypothesis, it follows that $Sy\in G$. Hence $Sy\in G$ as desired.
To conclude, if $y \not= Sx$ for each $x\in N$ then only $y$ must be equal to $0$.
And in the next proof is where I have troubles to close the induction because I only assume P1 and P3 and no more. I don't know how to prove it without the use of P2. Here is my approach:
Proof of P5: Let G be the set such that:
$$z\in G \iff z\in N \text{ and } z\not= Sz. $$
Clearly $0\in G$ as an specific case of P1. Suppose that $y\in G$ we need to show that $Sy\in G$ as well. We see that $y\not= Sy$ (as $y\in G$) just we need to prove that $Sy \not=S(Sy)$
And here exactly is where I'm stuck if we use P2 for almost trivial reason the claim holds. But I can't see how with this restriction I can close the induction. I would appreciate some help. Thanks in advance.
Let $N=\{0,1,2\}$ and define $S0=1,\ S1=S2=2.$ Then axioms $P_1$ and $P_3$ hold, but the statement $P_5$ does not.
So there can be no proof of $P_5$ using only $P_1,P_3.$