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How to prove that for every ordinal $\alpha$ there exist limit ordinal $\beta$ ,such that$\alpha\in\beta$ ?

Thank you.

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Take $\beta=\alpha+\omega$, where the sum is ordinal addition. –  Brian M. Scott Jan 9 '13 at 1:35

1 Answer 1

up vote 2 down vote accepted

You have several options.

  1. If you already proved basic things about ordinal addition, $\alpha+\omega$.
  2. If you have proved basic things about cardinals, $\alpha^+$.
  3. If you have to write full proof, show that there is a definable function from $\omega$ which sends $n$ to the $n$-th successor of $\alpha$. Use replacement and union to prove the existence of $\alpha+\omega$.

Also related: LIM is cofinal in ON

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