query: How could I prove $\omega_1$ is an uncountable ordinal without appealing to Foundation?
positive_passages.docid: 16971
positive_passages.text: Prove that no ordinal is an element of itself
negative_passages.docid: 126569
negative_passages.text: Can we find a well ordering on an infinite set with no largest element?
negative_passages.docid: 60004
negative_passages.text: Why is $\omega$ the smallest $\infty$?
negative_passages.docid: 19414
negative_passages.text: Uncountable subset with uncountable complement, without the Axiom of Choice
negative_passages.docid: 40937
negative_passages.text: Is the set of partial order relations over $ \mathbb{N}$ countable?
negative_passages.docid: 154230
negative_passages.text: In $\mathsf{ZF}$, for each set $A$ exists an ordinal $\alpha$ such that no surjection $A\to \alpha$ exists
negative_passages.docid: 53399
negative_passages.text: Proof of the uncountability of reals using the diagonal argument—problem?
negative_passages.docid: 149822
negative_passages.text: If $\omega \in \Omega^q(M)$, and $(\omega|_{U})_p = 0$, is $\omega_p = 0$?
negative_passages.docid: 119886
negative_passages.text: The cantor set is uncountable
negative_passages.docid: 83116
negative_passages.text: Uncountability of basis of $\mathbb R^{\mathbb N}$
negative_passages.docid: 42849
negative_passages.text: Is it true that $\omega=\{0,1,2,3,\ldots\}$ in ZFC?
negative_passages.docid: 43598
negative_passages.text: Do $\omega^\omega=2^{\aleph_0}=\aleph_1$?
negative_passages.docid: 116501
negative_passages.text: How to show equinumerosity of  the powerset of $A$ and the set of functions from $A$ to $\{0,1\}$ without cardinal arithmetic?
negative_passages.docid: 63115
negative_passages.text: If $\alpha$ is an indecomposable ordinal, why $\Gamma(\alpha\times{\alpha})=\alpha$?
negative_passages.docid: 157513
negative_passages.text: How can I prove that these sets are countable infinite?
negative_passages.docid: 146560
negative_passages.text: Is there any uncountably infinite set that does not generate the reals?
negative_passages.docid: 19924
negative_passages.text: Uncountable ordinals without power set axiom