query: Is the Axiom of Choice required to prove $|\alpha|=|L_\alpha|$, for infinite ordinals?
positive_passages.docid: 21881
positive_passages.text: Prove that $\forall\alpha\geq\omega$, $|L_\alpha|=|\alpha|$ without AC
negative_passages.docid: 157490
negative_passages.text: Statements equivalent to the Axiom of Choice
negative_passages.docid: 27899
negative_passages.text: Can we prove the existence of $A\cup B$ without the union axiom?
negative_passages.docid: 48278
negative_passages.text: Inequality about infinite cardinals
negative_passages.docid: 82739
negative_passages.text: Why is $\alpha \rightarrow \forall x(\alpha)$ not generally correct in first-order logic?
negative_passages.docid: 59187
negative_passages.text: Proof, is $\forall x : (P(x) \lor Q(x)) \Leftrightarrow \forall x : P(x) \lor \forall x : Q(x)$?
negative_passages.docid: 142219
negative_passages.text: Does the splitting lemma hold without the axiom of choice?
negative_passages.docid: 7214
negative_passages.text: Does linear ordering need the Axiom of Choice?
negative_passages.docid: 38726
negative_passages.text: What set theory axioms do I need to believe in uncountable ordinals?
negative_passages.docid: 39082
negative_passages.text: Proving $x^\alpha-\alpha x \le 1- \alpha $
negative_passages.docid: 59724
negative_passages.text: Proving that $x_n\to L$ implies $|x_n|\to |L|$, and what about the converse?
negative_passages.docid: 5035
negative_passages.text: Can unprovability unprovable? Is there an $\omega$-fold unprovability?
negative_passages.docid: 133870
negative_passages.text: if $\alpha\leq\beta$ then there is $\gamma$ such as $\alpha+\gamma=\beta$
negative_passages.docid: 120311
negative_passages.text: Equivalence of Axiom of Regularity
negative_passages.docid: 26161
negative_passages.text: The statement "$\alpha$ is a cardinal" is not absolute
negative_passages.docid: 99633
negative_passages.text: Let $α$ be an ordinal and $A$ be a set of ordinals. Then $\sup\limits_{β∈A} (α+β) = α+\sup\limits_{β∈A}(β)$
negative_passages.docid: 73526
negative_passages.text: The equivalence of "Every surjection has a right inverse" and the Axiom of Choice