 Models of this formula are, up to isomorphism, of the form \((M,\in,N,g,a)\), where \(\P(N)=M\), \(N\cap\P(N)=\emptyset\) and \(N\) is countably infinite. Thus the models are necessarily uncountable. This shows that second-order logic, unlike first order logic, has sentences with models but only uncountable ones. Thus the Downward Löwenheim-Skolem Theorem fails for second-order logic, as we already noted above. Combining (\ref{papow}) with the observations in  §5.1  yields the following result (by validity we mean the set of Gödel numbers of valid formulas):