# Tagged Questions

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### Show that an ordinal is a limit ordinal if and only if it is $\omega\cdot\beta$ for some $\beta$.

Show that an ordinal is a limit ordinal if and only if it is $\omega\cdot\beta$ for some $\beta$. To show the first implication, I was trying to use transfinite induction on beta to show that ...
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### Do we have $\overline{\bigcup\{A_\beta: \beta<\alpha\}}=\bigcup\{ \overline{A_\beta}: \beta<\alpha\}$

Let $\alpha$ be any ordinal, and for the family $\{A_\beta: \beta< \alpha\}$, we have $A_{\beta_1}\subset A_{\beta_2}$ when $\beta_1<\beta_2<\alpha$, then do we have ...
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### Proving of existence of limit ordinal

How to prove that for every ordinal $\alpha$ there exist limit ordinal $\beta$ ,such that$\alpha\in\beta$ ? Thank you.
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### Proving that an initial infinite ordinal is a limit ordinal

As in the title, An initial infinite ordinal is a limit ordinal. How can I start proving that?
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### How do I show there isn't an order isomorphism b/w the two sets $\{1, 2, 3,…\}$ and $\{1, 2, 3, …, \omega \}$

That is, how can I prove there isn't a bijection $f$ from one set to the other such that $f(x) < f(y)$ iff $x < y$?
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### Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$

This is an exercise from Kunen's book. Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$ are equal. What I've tried: I want to prove by using induction on $m$. ...
### Show that if $\kappa$ is an uncountable cardinal, then $\kappa$ is an epsilon number
Firstly, I give the definition of the epsilon number: $\alpha$ is called an epsilon number iff $\omega^\alpha=\alpha$. Show that if $\kappa$ is an uncountable cardinal, then $\kappa$ is an ...