# Other ways of proving that the set of all countable ordinals is uncountable

I know that the standard way of proving that the set of all countable ordinals is uncountable is by stating that if the set is countable, then it incurs Burali-Forti paradox.

Is there other ways of proving this?

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Not Russell's paradox so much as Burali-Forti's paradox, but they're closely related. –  Zhen Lin Oct 18 '12 at 9:43
Some proofs are given in this question: Uncountability of countable ordinals. (And you might want to have a look at linked questions, too.) –  Martin Sleziak Oct 18 '12 at 10:20

The set of all countable ordinals is the supremum of all countable ordinals, which is simply their unions. If $\omega_1$ were countable, $\omega_1+1$ would be countable too and hence $\omega_1+1<\omega_1$. Hence $\omega_1\in\omega_1+1\in\omega_1$, which shows that the set $\{\omega_1,\omega_1+1\}$ has no $\in$-minimum, which is impossible since the ordinals are well-ordered by $\in$.

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Wait, how is $\omega_1+1 \leq \omega$? Isn't $\omega$ the first infinite countable ordinal? –  DOT Oct 18 '12 at 10:20
@DOT Yes. But this is a proof by contradiction. Since the set of all countable ordinals is the supremum of all countable ordinals, and $\omega_1$ being countable implies $\omega_1+1$ we have $\omega_1+1<\omega_1$, which is by definition of the ordering on ordinals $\omega_1+1\in\omega_1$. –  Michael Greinecker Oct 18 '12 at 10:28
@CarlMummert Thank you. –  Michael Greinecker Oct 18 '12 at 11:25
I think that the contradiction has nothing to with the the fact those are ordinals; rather with the fact that the axiom of regularity fails. –  Asaf Karagila Oct 18 '12 at 13:30
@Asaf But the proof works even when the axiom of regularity fails. –  Michael Greinecker Oct 18 '12 at 19:50

I think that all proofs will be similar in some way to the Burali-Forti paradox. Here is a proof that is slightly different than other proofs on this site, so maybe someone will find it useful. Note that the Axiom of Regularity is not used anywhere.

Definition: An ordinal is a transitive set well-ordered by $\in$.

Fact: The class of ordinals is transitive and well-ordered by $\in$.

Definition: $\omega_1$ is the class of countable ordinals.

Fact: $\omega_1$ is a set.

To see that $\omega_1$ is an ordinal using the above facts, it remains to observe that every element of a countable ordinal is a subset of that ordinal and is therefore countable also.

Now suppose toward a contradiction that $\omega_1$ is countable. Then $\omega_1 \in \omega_1$ by definition. This contradicts the fact that the class of ordinals is well-ordered by $\in$.

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