First-order logic formula satisfiable only if the domain of the valuation is infinite?

pooj4

New member
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

Well-known member
MHB Math Scholar
$$\displaystyle (\forall x\forall y\,(S(x)=S(y)\to x=y))\land \forall x\,S(x)\ne0$$

pooj4

New member
$$\displaystyle (\forall x\forall y\,(S(x)=S(y)\to x=y))\land \forall x\,S(x)\ne0$$
thanks that helps