query: Is it possible that A ∪ B ⊢ α ∧ ¬α, but there be no β such that A ⊢ β and B ⊢ ¬β?
positive_passages.docid: 155679
positive_passages.text: If union of two theories is not consistent one of them implies a sentence which the other implies opposite of it
negative_passages.docid: 66563
negative_passages.text: Is $ (A∪B)×(C∩D)=(A∪C)×(B∪D)$ true for all sets A,B,C and D?
negative_passages.docid: 111723
negative_passages.text: Can it be proven in $\sf ZF^{\neg\infty}$ that the sets $x_n=\{x_{n+1}\}$ do not exist?
negative_passages.docid: 120517
negative_passages.text: Is it possible to prove that a particular statement cannot be disproved without creating a contradiction?
negative_passages.docid: 101453
negative_passages.text: Existence of a Boolean algebra with a unique ultrafilter in ZF
negative_passages.docid: 32180
negative_passages.text: Why is "$A$ unless $B$" equivalent to $A \lor B$?
negative_passages.docid: 133870
negative_passages.text: if $\alpha\leq\beta$ then there is $\gamma$ such as $\alpha+\gamma=\beta$
negative_passages.docid: 77859
negative_passages.text: Prove or find a counter example to the claim that for all sets A,B,C if A ∩ B = B ∩ C = A ∩ C = Ø then A∩B∩C ≠ Ø
negative_passages.docid: 142168
negative_passages.text: can we a prove ⊢ (α → α) → (α → α)
negative_passages.docid: 113735
negative_passages.text: Do we need to prove that if x∉A in proposition "if A⊆B then A⋃B=B"?
negative_passages.docid: 101741
negative_passages.text: Selection of logical connectives {¬,∨,∧,⇒,⇔} in set theory?
negative_passages.docid: 21229
negative_passages.text: If an element is not in the set difference $A\setminus B \: \:?$
negative_passages.docid: 4700
negative_passages.text: Can paraconsistent or other logics make the impossible happen?
negative_passages.docid: 28539
negative_passages.text: Does any uncountable set contain two disjoint uncountable sets?
negative_passages.docid: 88982
negative_passages.text: Prove that for any two sets $A$ and $B$, $A\notin B$ or $B\notin A$
negative_passages.docid: 139841
negative_passages.text: Proving $A\setminus(B\setminus C)$ is not equal to $(A\setminus B)\setminus C$
negative_passages.docid: 93557
negative_passages.text: Intuitively, why does (B → S) ≡ (¬B ∨ S)?