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Cantor's construction of $\aleph_1$ is to notice that all ordinals constructed after $\omega$ are countable, take the union of all countable ordinals, and then show that this union can not be countable. However, in ZFC the union can only be taken over a set. I can see how to relate countable ordinals to elements of $2^{\aleph_0}$ proving that their collection is a set by specification, but that requires the power set axiom. The "slicker" modern way, declaring $\aleph_1$ to be the least uncountable cardinal, already presupposes the existence of uncountable cardinals. That's even worse since in addition to power set one needs choice to prove that $2^{\aleph_0}$ is an aleph, and then it is greater by the diagonal argument.

It wouldn't surprise me that some voodoo with collection or replacemnt schemas does the trick, but then my question is what does it translate to in "naive" terms of Cantor? What reason do we have for the collection of countable ordinals to be a set (without invoking formulas in the first order logic and definable functions)? If power set isn't required why can't Cantor's argument be reproduced to create inaccessible cardinals by noticing that all cardinals produced after $\aleph_0$ are, by construction, accessible?

Conifold
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  • I've always thought $\aleph_1$ was defined as the cardinality of the set of all countable ordinals. That it's the least uncountable cardinal might fail if AC is false, since there might be other uncountable cardinals not comparable with it. – Michael Hardy May 24 '14 at 22:12
  • Regarding to aspects involving the axiom of choice, http://math.stackexchange.com/questions/46833/how-do-we-know-an-aleph-1-exists-at-all/ has a discussion on the topic. – Asaf Karagila May 24 '14 at 22:23
  • (In short, without the axiom of choice, but using power set axiom and replacement axioms, we can show that $\omega_1$ can be produced by a definable function whose domain is $\mathcal P(\omega)$; or more concrete there is a definable linear order of $\cal P(P(\omega))$ of order type $\omega_1$, so separation we can isolate that set and then collapse it to be a von Neumann ordinal, if one is so inclined.) – Asaf Karagila May 24 '14 at 22:26
  • As for the final remark. There's no voodoo there. You just need to study a bit to understand that the existence of $\aleph_0$ follows only from the axiom of infinity. This understanding reflects why we cannot prove the existence of inaccessible cardinals without stronger axioms of infinity; and why all the cardinals which can be proved to exist using just the axioms of $\sf ZFC$ are necessarily accessible (or $\aleph_0$). – Asaf Karagila May 24 '14 at 22:28
  • In his original paper on ordinals Cantor asserts three "generation principles" for them. The first one is taking a successor, the second one he uses to create omega, taking the union over all generated successors. Axiom of infinity later justified this by embedding all successors into a pre-existent set. But a pre-existent set is missing for omega1, without power sets. Does replacement compensate for that? If so, why does it work for countability but not for accessibility? "Third principle" would work for both. Any suggestions on changing the question, I am new here. – Conifold May 24 '14 at 23:20

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