I'm looking for a set containing an element 0 and a successor function s that satisfies the first two Peano postulates (s is injective and 0 is not in its image), but not the third (the one about induction). This is of course exercise 1.4.9 in MacLane's Algebra book, so it's more or less homework, so if you could do the thing where you like point me in the right direction without giving it all away that'd be great. Thanks!
Since your set has 0 and a successor function, it must contain $\Bbb N$. The induction axiom is what ensures that every element is reachable from 0. So throw in some extra non-$\Bbb N$ elements that are not reachable from 0 and give them successors. There are several ways to do this.
Geometrically, $\Bbb N$ is a ray with its endpoint at 0. The Peano axioms force it to be this shape. Each axiom prevents a different pathology. For example, the axiom $Sn\ne 0$ is required to prevent the ray from curling up into a circle. It's a really good exercise to draw various pathological shapes and then see which ones are ruled out by which axioms, and conversely, for each axiom, to produce a pathology which is ruled out by that axiom.
Addendum: I just happened to be reading Frege's Theorem and the Peano Postulates by G. Boolos, and on p.318 it presents a variation of this exercise that you might enjoy. Boolos states a version of the Peano axioms:
And then says:
Your job: find the models!
Think about the question for a moment (and think about the first two parts I'm assuming you got).
You need some set X with a unary operation and an element 0 that satisfies P.P. (i) and (ii).
When you constructed the first two sets, was your unary operation x+1? Or did you create some other mapping? (It shouldn't have been x+1 given that, for one of them at least, you needed 0 in the image).
Can you create a unary operation over N that is injective and does not include 0 in its image, but whose image is not N?