Proving the Overspill Principle. The Overspill Principle, as I have encountered it, states: > Given $M$ a non-standard (i.e. not isomorphic to the naturals) model of Peano Arithmetic, $\varphi$ a formula with $n+1$ free variables and $x_1, \ldots x_n \in M$. If in $M$ we have that $\varphi(n, x_1,\ldots, x_n)$ holds for all $n$, then there is an infinite $x\in M$ such that $\varphi(x,x_1,\ldots, x_n)$ holds. I do not know how can one see this. My first thought was to apply compactness somehow but I am not so sure now.

Suppose otherwise, namely that $\varphi(n,x_1,\ldots,x_n)$ holds for every numeral $n$, but does not hold with any non-standard element in the model.

Then both of $$ \varphi(0,x_1,\ldots,x_n) \\\ \forall y: \varphi(y,x_1,\ldots,x_n) \to \varphi(y+1,x_1,\ldots,x_n) $$ are true in $M$, so by the induction axiom we should have $$ M\models \forall x: \varphi(x,x_1,\ldots,x_n) $$ but that contradicts the assumption that $M$ contains a non-standard element that $\varphi$ is false for!