math_proof Group Title Peano's axioms prove 2 years ago 2 years ago

1. math_proof

Starting from Peano's axioms prove that if $n \in \mathbb{Z}^+ and n \ne1$ then n is a successor, i.e. s(a)=n for some $a \in \mathbb{Z}^+$

2. math_proof

[Let A=Im(s) U {1} and prove that A=Z+]

3. KingGeorge

I think the first could happen through induction. Base case: If $$n=2$$, then $$s(1)=2$$, so we're good. Now we assume up to some $$k<n$$. Now we need to show it's true for $$k+1$$. But since $$s(k)=k+1$$ for $$k\in\mathbb{Z}^+$$, we're done. I've got to go for a bit now. I'll come back to the second part later.

4. math_proof

which second part? whats in thee brackets is the hint

5. KingGeorge

If that's the hint, then we're completely done. For $$n\in\mathbb{Z}_{\ge 2}$$, we have that $$n\in \text{im}(s)$$, and since $$\text{im}(s)\subseteq \mathbb{Z}^+$$ by definition, we've proved that $$A=\mathbb{Z}_{\ge2}$$, so $$A\cup \{1\}=\mathbb{Z}^+$$.