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Let $\Lambda$ be the set of all countable limit ordinals. Does there exist an injective function $f:\Lambda\to\omega_1$ with the properties:

  1. $\forall \lambda\in\Lambda:~f(\lambda)<\lambda$
  2. $\forall\alpha<\omega_1~~\exists\beta<\omega_1~~\forall\lambda>\beta:~f(\lambda)>\alpha$ ?
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    $\begingroup$ Though the answer below answers your question, you might be interested to know that an injective function $f:\Lambda\to\omega_1$ automatically satisfies $2$: for any $\gamma\leq\alpha$ there is at most one ordinal $a_\gamma$ such that $f(a_\gamma)=\gamma$. Then $\beta=\sup_{\gamma\leq\alpha}a_\gamma$ will work. $\endgroup$
    – Wojowu
    Commented Apr 24, 2019 at 8:25

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No. The first property is known as $f$ being regressive. Fodor’s Lemma says that any regressive function on a stationary set is constant on a stationary subset. In particular, because $\Lambda$ is club (and thus stationary), such $f$ cannot be injective.

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