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The only reasoning I've seen given for this is that it's uncountable because it can't include itself an element. I'm a little unconvinced and was looking for a more proper formal proof demonstrating the equality: $\omega_1 = \aleph_1$.

Acid2
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    I feel that I may have answered this once or twice before on this site. – Asaf Karagila Sep 04 '12 at 12:41
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    $\omega_1$ is usually defined to be the the least uncountable cardinal. So it is uncountable by definition. Thus every ordinal in $\omega_1$ is countable. Moreover any countable ordinal $\alpha$ cannot be larger than or equal to $\omega_1$ and so $\alpha \in \omega_1$. Thus $\omega_1$ is the set of countable ordinals. – Levon Haykazyan Sep 04 '12 at 12:42
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    See: http://math.stackexchange.com/questions/46833/how-do-we-know-an-aleph-1-exists-at-all/46836#46836 and http://math.stackexchange.com/questions/71726/uncountability-of-countable-ordinals/71733#71733 – Asaf Karagila Sep 04 '12 at 12:54
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    Your argument is fine. As you say, if $\omega_1$ is countable, then it belongs to the set of all countable ordinals, so $\omega_1\in\omega_1$. This contradicts that ordinals are well-ordered. The reason why you may be a little unconvinced is that you have not yet verified that there is a set of all countable ordinals, to begin with. (If there is such a set, it should not be hard for you to check that it is indeed an ordinal.) Once you argue that the collection of countable ordinals indeed forms a set, we are done. – Andrés E. Caicedo Oct 01 '14 at 01:45

3 Answers3

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Following Henning's suggestion, posting my comment as an answer.

$\omega_1$ is usually defined to be the the least uncountable ordinal number. So it is uncountable by definition. Thus every ordinal in $\omega_1$ is countable. Moreover any countable ordinal $\alpha$ cannot be larger than or equal to $\omega_1$ and so $\alpha \in \omega_1$. Thus $\omega_1$ is the set of countable ordinals.

chharvey
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Levon Haykazyan
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    I'm a little confused. If $\omega_1$ is defined to be the least uncountable cardinal, I can't see/understand exactly how that gives you as a consequence that every ordinal in $\omega_1$ is countable (you used the word 'thus'). Can you please be more clear? I assume you're not using the hypothesis in the question there that every countable ordinal is in $\omega_1$ because then it becomes circular reasoning to prove the hypothesis using the the hypothesis. – Acid2 Sep 04 '12 at 18:06
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    @Bakhtiar: For ordinals it holds that $\alpha<\beta\iff\alpha\in\beta$. If $\omega_1$ is the first uncountable ordinal it means that $\alpha<\omega_1$ implies that $\alpha$ is not uncountable (else it would be the first, preceding $\omega_1$). – Asaf Karagila Sep 05 '12 at 00:12
  • @AsafKaragila: Okay, I see, we're viewing the least uncountable cardinal as an ordinal (the first uncountable ordinal) and then it follows every ordinal in it ($\omega_1$) is countable... makes sense. Thanks. – Acid2 Sep 05 '12 at 12:54
  • @Levon I believe your second sentence should read, "$\omega_1$ is usually defined to be the least uncountable ordinal." Making this edit would clear up some confusion in the comments. As a matter of fact, $\omega_1$ technically isn't a cardinal number. When referring to the least uncountable cardinal, mathematicians typically use $\aleph_1$. – chharvey Sep 30 '14 at 20:27
  • @chharvey: While it's always good to correct and improve answers, bumping something from two years ago which had a minor issue that was clarified in the comments, and there are only three comments to begin with (which means that all the comments are very visible)... seems a bit of overreaching here. – Asaf Karagila Sep 30 '14 at 20:55
  • Also, @chharvey, as a matter of fact, $\omega_1$ is a cardinal number if we define cardinals as initial ordinals. But I agree that it's good to make the distinction between ordinals and cardinals, if only for the sake of ordinal and cardinal arithmetics. – Asaf Karagila Sep 30 '14 at 20:56
  • @AsafKaragila You're right, I just read that cardinals are defined as ordinals after posting my comment... ::foot in mouth:: Still, making the edit would help clear the confusion. However I disagree that it is "overreaching." In my opinion, a question that can be improved should, no matter how old, especially for users who so happen to stumble upon it years later. – chharvey Oct 03 '14 at 02:44
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The reasoning is correct: The set $\omega_1$ of all countable ordinals (here always meant as: including finite ones) is an ordinal, hence we have $\omega_1\notin\omega_1$ (even in set theories that allow Quine atoms), hence it is not a countable ordinal. Since it is definitely an ordinal, it must be an uncountable ordinal.

EDIT: removed bad reason for $\le$ argument

Hagen von Eitzen
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Why is this not a reasonable reason?

Ordinals are sets which are transitive and well-ordered by $\in$. $\omega$ is defined to be the first infinite cardinal, $\omega_1$ is defined to be the first ordinal which is larger than $\omega$ not in bijection with any of its members.

The definition of $\omega_1$, then, implies that $(1)$ There is no bijection between $\omega_1$ and $\omega$, that is to say that $\omega_1$ is uncountable; and $(2)$ that every ordinal below $\omega_1$ has to be countable. Since $\in$ is well-founded we have that $\omega_1\notin\omega_1$.

Asaf Karagila
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