query: How do we know that $\omega_1$ exists in ZF?
positive_passages.docid: 28764
positive_passages.text: How do we know an $ \aleph_1 $ exists at all?
negative_passages.docid: 5584
negative_passages.text: How strong is this second-order version of ZFC?
negative_passages.docid: 101453
negative_passages.text: Existence of a Boolean algebra with a unique ultrafilter in ZF
negative_passages.docid: 40114
negative_passages.text: Factor into irreducibles in $\mathbb{Z}[\omega]$ (Eisenstein Integers).
negative_passages.docid: 12469
negative_passages.text: $\varphi$ in $\operatorname{Hom}{(S^1, S^1)}$ are of the form $z^n$
negative_passages.docid: 93621
negative_passages.text: Prove that $\mathbb{Z}[\omega]$ is an integral domain
negative_passages.docid: 96355
negative_passages.text: What is the first order axiom characterizing a field having characteristic zero?
negative_passages.docid: 30587
negative_passages.text: Uncountable well ordered set in $Z^-$ theory.
negative_passages.docid: 127200
negative_passages.text: Are there fields and ordered fields of every infinite cardinality?
negative_passages.docid: 975
negative_passages.text: Is there a known well ordering of the reals?
negative_passages.docid: 127622
negative_passages.text: Does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?
negative_passages.docid: 49155
negative_passages.text: $f: \Omega \rightarrow \Omega$ holomorphic, $f(0) = 0$, $f'(0) = 1$ implies $f(z) = z$
negative_passages.docid: 139334
negative_passages.text: Determining the existence or non-existence of Non Trivial Ideals of $L(V)$
negative_passages.docid: 58602
negative_passages.text: In ZF, how would the structure of the cardinal numbers change by adopting this definition of cardinality?
negative_passages.docid: 78290
negative_passages.text: Countable axiom of choice: why you can't prove it from just ZF
negative_passages.docid: 112008
negative_passages.text: Ordinal $\omega_1$ is not compact
negative_passages.docid: 2667
negative_passages.text: Given $S \hookrightarrow T$ construct $U ≈ T$ disjoint from $S$ in Z set theory?