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 ?