The sentence schema you're given is the definition of the principle of finite induction. It should be sufficient to derive the other axioms from the first two and induction.
Let be the set of sentences consisting of S1, S2, and all sentences of the form
where is a wff (in the language of ) in which no variable except occurs free. Show that . Conclude that . (Here is by definition . The sentence displayed above is called the induction axiom for .)
The necessary definitions can be found here (especially, page 2, slide 7).
We show that if , then . Therefore, we show each is a sentence in . If (resp. , by assumption.
S3 asserts that any nonzero number is the successor of somthing.
How do I show that each S3, S4.1, S4.2, ... , and S4.n is a sentence in ?
From there, how do I conclude ?
If T is a set of sentences, is Cn T the set of all sentences that can be derived from T?
Take the property "this object is either zero, or is the successor of some number x". You would agree this is S3 I hope. Well, induction is easy:
1) Zero meets this property. (base)
2) For any number y, if y meets this property, then Sy meets this property. (inductive step)
3) By FPI, all objects meet this property.
You're allowed to assume all sentences of the form:
Before we begin the induction, we define and plug it into the sentence for .
Using the above sentence, we can prove holds for zero. We can also prove that if it holds for some number, it holds for the successor of that number (because of how we defined this is always true). Based on the expression above, this means : every number is either zero or a successor. But this is exactly what we wished to show.
Does that help? :) Try the others on your own if you get it now.
For a procedure to create a proof by induction, see this post.