The signature consists of $S$, $0$ & $<$ and the axioms are:
I - $\forall x (S(x) \not= 0)$
II - $\forall x \forall y (S(x) = S(y) \to x = y)$
III - First-order Induction schema
IV - $<$ is a linear order
V - $\forall x (x < S(x))$
Are the above axioms enough to prove that $\forall x (\neg \exists k (x < k < S(x))$ i.e. there is no natural number $k$ between $x$ and $S(x)$?
