query: A model of geometry with the negation of Pasch’s axiom?
positive_passages.docid: 154961
positive_passages.text: Are there simple models of Euclid's postulates that violate Pasch's theorem or Pasch's axiom?
negative_passages.docid: 102899
negative_passages.text: Category theory with objects as logical expressions over $\{\vee,\wedge,\neg\}$ and morphisms as?
negative_passages.docid: 75225
negative_passages.text: Exercise $ 3.4.15 $ of David Marker’s “Model Theory”.
negative_passages.docid: 84378
negative_passages.text: Does axiom of foundation/regularity protect against Russell-like paradoxes?
negative_passages.docid: 127826
negative_passages.text: Is the geometric realization of a pointed category contractible?
negative_passages.docid: 135222
negative_passages.text: What are the consequences if Axiom of Infinity is negated?
negative_passages.docid: 150882
negative_passages.text: If $\Gamma \cup \{ \neg \varphi \}$ is inconsistent, then $\Gamma \vdash \varphi$
negative_passages.docid: 65308
negative_passages.text: Axiom of Choice: Family of non-empty sets mutually disjoint or not?
negative_passages.docid: 141299
negative_passages.text: Alternative set theories
negative_passages.docid: 83230
negative_passages.text: A useful illustration for differential geometry
negative_passages.docid: 147794
negative_passages.text: Applications of Category Theory Outside of Mathematics (the discipline)
negative_passages.docid: 48495
negative_passages.text: Is it possible to prove the independence of the parallel postulate without non-euclidean geometry?
negative_passages.docid: 151215
negative_passages.text: Paradox of General Comprehension in Set Theory, other than Russell's Paradox
negative_passages.docid: 31389
negative_passages.text: consistency of ZF\{Power Set axiom}
negative_passages.docid: 102202
negative_passages.text: The smallest infinity and the axiom of choice
negative_passages.docid: 79727
negative_passages.text: Direct proof of non-flatness
negative_passages.docid: 110472
negative_passages.text: Existence of a Pivot Point in Euclidean Geometry