<P> We have a language L N T = (0, S) (\ displaystyle (\ mathfrak (L)) _ (NT) = \ (0, S \)) where 0 (\ displaystyle 0) is a constant symbol and S (\ displaystyle S) is a unary function and the following axioms: </P> <Ol> <Li> ∀ x . ¬ (S x = 0) (\ displaystyle \ forall x. \ lnot (Sx = 0)) </Li> <Li> ∀ x . ∀ y . (S x = S y → x = y) (\ displaystyle \ forall x. \ forall y. (Sx = Sy \ to x = y)) </Li> <Li> (φ (0) ∧ ∀ x . (φ (x) → φ (S x))) → ∀ x . φ (x) (\ displaystyle (\ phi (0) \ land \ forall x. \, (\ phi (x) \ to \ phi (Sx))) \ to \ forall x. \ phi (x)) for any L N T (\ displaystyle (\ mathfrak (L)) _ (NT)) formula φ (\ displaystyle \ phi) with one free variable . </Li> </Ol> <Li> ∀ x . ¬ (S x = 0) (\ displaystyle \ forall x. \ lnot (Sx = 0)) </Li> <Li> ∀ x . ∀ y . (S x = S y → x = y) (\ displaystyle \ forall x. \ forall y. (Sx = Sy \ to x = y)) </Li>

Define axiom postulate and theorem also write one example of each