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 theinduction axiomfor .)

The necessary definitions can be found here (especially, page 2, slide 7).

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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 ?