In my previous note, there was a typo in the quotation I copied
from Shoenfield's book. Following is the corrected version: (01)
> If A is a formula of [Peano's theory stated in first-order
> logic that contains a free variable x], then the formula
>
> A(0) & (For all x (A(x) -> A(S x))) -> A
>
> is called an _induction axiom_. The theory obtained from
> [Peano's axioms by omitting his second-order axiom] and
> adding all the induction axioms as new nonlogical axioms
> axioms is called _Peano Arithmetic_. (02)
My typo was "induction axioms as new induction axioms" instead
of just "induction axioms". That's why we need a simple way
to "cut and paste" from a printed book. (03)
John (04)
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