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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.

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    I'm not sure if I qualify as more fluent, but it seems okay to me. It could be perhaps made simpler if you skolemized $\kappa$ at the beginning (then you would have q.e. so all the steps (proofs of elementary inclusions, constructions of elementary extensions) would be completely automatic), but it's up to you whether or not to see it that way.2012-11-05

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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.