Existence of elementary substructures of a uncountable structure over a countable language Let $\kappa$ be an regular uncountable cardinal carrying a $\tau$-structure for some countable language $\tau$. What can be said regarding the existence of ordinals $\alpha <\kappa$ carrying elementary substructures of $\kappa$ ?
This is an (altered) question from an exercise which I am having difficulties solving. The Löwenheim–Skolem theorem doesn't seem to provide any direct insights here. 

Hint: Take an ordinal $\alpha_0<\kappa$, use it to generate an elementary submodel $M_0$; if its universe is an ordinal we are done. Otherwise take $\alpha_1=\sup|M_0|$; reiterate. Prove that $\alpha=\sup\alpha_n$ is an elementary submodel. 
