I have to give a talk which mentions the Peano Axioms for Arithmetic. I want to make sure when i am explaining them in English I am describing them correctly.
(PA1) $(\forall x_1)( \neg s(x_1) = 0)$ $\ \ $
(PA2) $(\forall x_1) (\forall x_2) (s(x_1) = s(x_2) \to x_1 = x_2)$
(PA3) $(\forall x_1) x_1 + 0 = x_1$
(PA4) $(\forall x_1) (\forall x_2) x_1 + s(x_2) = s(x_1 + x_2)$
(PA5) $(\forall x_1) x_1 * 0 = 0$
(PA6) $(\forall x_1) (\forall x_2) x_1 s(x_2) = x_1 * x_2+x_1$
(PA7) $\phi (0) \to ((\forall x_1) (\phi (x_1) \to \phi(s(x_1))) \to (\forall x_1)\phi(x_1)$
(PA1) says 0 is not the succesor of any number.
(PA2) says Two numbers of which the successors are equal are themselves equal
PA3 says that additive identity exists
PA4 says successor function ()
PA5 says there exists a multiplicative identity
PA7 is the induction axiom and says If a set of numbers contains zero and also the successor of every number in the set, then every number is in the set.
Any help will be appreciated