Infinite Natural Numbers: First-Order Logic Formula Explained

[pooj4]

There are an infinite number of natural numbers. Why is that? Well this follows from the following facts:

(i) There is at least one natural number.

(ii) For each natural number there is a distinct number which is its successor, i.e., for each number $x$ there is a distinct number $y$ such that $y$ stands in the
successor relation to $x$.

(iii) No two natural numbers have the same successor.

(iv) There is a natural number, namely 0, that is not the successor of any number.Bearing these facts in mind, what's a formula in first-order logic that that is satisfiable by a valuation only if the domain of the valuation is infinite. Contain some non-logical vocabulary in presentation of course.

 

[Evgeny.Makarov]

$$(\forall x\forall y\,(S(x)=S(y)\to x=y))\land \forall x\,S(x)\ne0$$

 

[pooj4]

$$(\forall x\forall y\,(S(x)=S(y)\to x=y))\land \forall x\,S(x)\ne0$$

thanks that helps