query: Show that $\Sigma \vdash\varphi$ if and only if $\Sigma\,\cup \{\neg\varphi\}$ is inconsistent.
positive_passages.docid: 150882
positive_passages.text: If $\Gamma \cup \{ \neg \varphi \}$ is inconsistent, then $\Gamma \vdash \varphi$
negative_passages.docid: 84251
negative_passages.text: Show that $\Gamma \cup \{\neg \phi\}$ is satisfiable if and only if $\Gamma\not \models \phi$
negative_passages.docid: 105303
negative_passages.text: 'Obvious' theorems that are actually false
negative_passages.docid: 40048
negative_passages.text: Finding the fallacy in this broken proof
negative_passages.docid: 5035
negative_passages.text: Can unprovability unprovable? Is there an $\omega$-fold unprovability?
negative_passages.docid: 65379
negative_passages.text: Deriving $\neg R$ from $\{R↔(R∨(P∧¬P)), R↔¬P, ¬P→(P↔(Q→Q)), P→Q\}$
negative_passages.docid: 150385
negative_passages.text: The use of the Axiom of Choice in this proof of the sequence lemma
negative_passages.docid: 139622
negative_passages.text: Generator of rational functions unchanged under $\sigma(X) = X + 1$
negative_passages.docid: 120442
negative_passages.text: Prove Borel sigma-algebra translation invariant
negative_passages.docid: 11710
negative_passages.text: How to show that $\mathbb{Q}$ is not $G_\delta$
negative_passages.docid: 26369
negative_passages.text: Proving seemingly obvious equations with measure theory
negative_passages.docid: 52014
negative_passages.text: Showing $(C[0,1], d_1)$ is not a complete metric space
negative_passages.docid: 106799
negative_passages.text: Proof Verification: Converse of Intermediate Value Theorem
negative_passages.docid: 38946
negative_passages.text: Show that $\{\cup_{n\in K} (n, n+1]: K \subset \mathbb{Z}\}$ is a $\sigma$-algebra on $\mathbb{R}$
negative_passages.docid: 80920
negative_passages.text: Prove or disprove: For any $v\in\mathbb{R}^n, \|v\|_1\|v\|_\infty \leq \frac{1+\sqrt{n}}{2}\|v\|_2^2$
negative_passages.docid: 62936
negative_passages.text: Proving equivalencies of two versions of the Axiom of Choice
negative_passages.docid: 117185
negative_passages.text: The union of a strictly increasing sequence of $\sigma$-algebras is not a $\sigma$-algebra