# Tag: Open Lab #6

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### Logic on Math StackExchange

- The Error-Correction Procedure calculation in AI? February 25, 2024I have the following homework (AI-related) with solution: There is a training set is in the form of (s1, ..., s8, d), where the first 8 elements are the Pacman’s sensor readings, and d is the label of this input, with d = 1 meaning a positive, and 0 a negative example. Now ,I need […]oscar
- Is the Axiom of Completeness logically equivalent to "There is no proper superset of $\mathbb R$ that is an ordered Archimedean field"? February 25, 2024The Axiom of Completeness can be formulated as: There exists a set $R$ such that: $R$ is an ordered Archimedean field Any nonempty subset of $R$ with an upper bound has a least upper bound. Recently, I read something that suggested this is logically equivalent to the following: There exists a set $R$ such that: […]SRobertJames
- Property Equivalent to Maximally Consistent February 24, 2024This is a question about Gentzen calculus. $X$ is a set of formulas with the symbols $\{\lnot, \land\}$, and $a$ is such a formula. On page 27 of "A Concise Introduction to Mathematical Logic" by Rautenberg, it states: it easily follows that $X$ is maximally consistent iff either $a \in X$ or $\lnot a \in […]Enrico Borba
- When mathematicians say "true" do they mean "true in all models"? February 24, 2024According to the comments to this question, Truth is ordinarily defined by reference to models. If so, even axioms and theorems are not true without reference to a model. However, when mathematicians say "this is true," they are usually not referring to a specific model. Is he saying "true in all models"?MathMan
- Correctly applying Disjunctive Syllogism February 24, 2024If one has $\neg p \vee q$ and $\neg q$, does one need to apply the Commutative Law to $\neg p \vee q$ to obtain $q \vee \neg p$, before concluding that $\neg p$ ?dingus
- Do all propositions in ZFC have truth values? [closed] February 24, 2024I wonder if all propositions in mathematics have a truth value. As an example, I ask about ZFC.MathMan
- Show that for positive integers x, y , z and w If x < y and z < w then zx < wy [duplicate] February 23, 2024My attempt at solving this goes $$z\lt w\Rightarrow z\bullet z\lt w\bullet w$$ since $z\lt w$ is logically equivalent to $x\lt y$ just with different symbols, then $$z\lt w \Rightarrow z\bullet x \lt w\bullet y$$ However, in the proof of the first theorem, I took for granted that $w\bullet z = z\bullet w$ which is not […]Zelalem Ewnetu
- Subset of integers called by a somewhat ill-defined property February 23, 2024I know there were some issues with set theory that involved self-reference and famous example being that of Russell's example. Here, I have a set that I use a somewhat vague term "use" to construct but eventually answer leads to a contradiction with the property itself. What I want to ask is, can someone explain […]Mahammad Yusifov
- Admissibility of Löb's rule in basic modal logic K February 23, 2024While I was preparing a talk on the admissible rules of modal logic, I found the following fact in Wikipedia (see https://en.wikipedia.org/wiki/Admissible_rule#Examples). It says that Löb's rule $(\square p \to p)/p$ is admissible in minimal modal logic K (а rule $\phi/\psi$ is called admissible in logic $L$, if for all substitution $\sigma$, such that $\vdash_L\sigma(\phi)$, […]lnv619
- HoTT and isomorphisms February 23, 2024I have heard that Homotopy Type Theory makes it so that isomorphic objects are “equal”. I wonder how this squares with a lot of mathematical examples from Algebra and Set Theory, where the nature of the isomorphism, or a certain class of isomorphisms, and how they interact with other morphisms, is relevant. How can you […]mbsq

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