Hi everyone,
I have a commitment tomorrow morning until 9:30, but I will be in my office (or in the classroom) after that. We’ll be doing exam review in class for the first half hour or so, so bring your questions!
See you tomorrow,
Prof. Reitz
Tag: office hours
Logic on Math StackExchange
- First order logic, why are the quantifier rules of inference reasonable? April 18, 2024This picture is from the book "Mathematical Logic, 2nd edition, Christopher C. Leary, Lars Kristensen" : I have two questions : Why are the quantifier rules of inference reasonable as they write in the start of the text ? Consider these variations : $$(\{\psi\rightarrow \phi\}, \psi \rightarrow (\exists x \phi)\})$$ $$(\{\phi\rightarrow \psi\}, (\forall x \phi) […]user394334
- Logical Definition of Isomorphism April 18, 2024I'm trying to define isomorphism in a logical way. Is the following statement true for Isomorphism's definition? Let $\langle S,⋆\rangle$ and $\langle S',⋆'\rangle$ be Algebraic Structures. These two structures are isomorphic if and only if: $$\left( \exists \phi : S \rightarrow S' \right) \left(\ \forall a,b \in S \right) \left[ \left( \phi (a⋆b) = \phi(a) […]Karaji
- Proof by contradiction square root of 3 is irrational [closed] April 18, 2024I think it requires circular thinking Please see this YouTube video https://www.youtube.com/watch?v=XQn47l2bt1g for details I found this alternative approach promising https://math.stackexchange.com/a/930492/1224828, and would like to know what is wrong with the approach in the YouTube video please.Caleb
- Is the inference rule another notation for $\implies$? April 18, 2024In mathematical logic, when we write a formula of the metalanguage (such as $A \vdash B$) above an horizontal rule and another formula (such as $\vdash A \to B$), is it the same as writing a meta implication sign $\implies$ between the two formulas?Anserin
- Prove that if a frame is dense then $\Box \Box A \to \Box A $ is valid [duplicate] April 18, 2024I am having trouble with the following question on modal logic. A modal logic frame $M =\langle W, R\rangle$ is dense whenever $$\forall x\forall y(Rxy\rightarrow\exists z(Rxz \land Rzy))$$ Prove that the formula $$\Box\Box A\to\Box A $$ is valid in every dense frame. We don't make any assumptions about the frames. How can this be done?A_alk
- Confusion on exercise on $(\mathbb Z, +)$ in Buechler's Stability theory book April 18, 2024Given as exercise 2.5.1: Show that the complete type realized by $1$ in $(\mathbb Z,+)$ is non-isolated. HINT Use the preceding exercise. The previous exercise: Let $\vec a$ and $\vec b $ be finite sequences from $\mathcal M.$ Prove that $\operatorname{tp}_{\mathcal M}(\vec a\vec b)$ is isolated iff $\operatorname{tp}_{\mathcal M}(\vec a/\vec b)$ and $\operatorname{tp}_{\mathcal M}(\vec b)$ […]spaceisdarkgreen
- Are there any problems about the difference between set theoretic definitions of polynomials? April 17, 2024I am a novice about this question, so if there is a misunderstanding then I apologize for it. As for Peano axioms, if I choose Zermelo natural numbers, and you choose von Neumann ones, then this doesn't cause any problems because we can prove Peano axioms are categorical i.e. it is guaranteed my model, and […]categoricalequivalent
- Application of the Löwenheim-Skolem Theorem for theories with uncountable signatures? April 17, 2024The Löwenheim-Skolem asserts the following: For a first order (L_(ω,ω)) theory with signature σ, every infinite σ - structure M and every infinite cardinal κ ≥ |σ|there is a σ-structure N such that |N| = κ and such that: if κ < |M| then N is an elementary substructure of M; if κ ≥ |M| […]SJe967
- Is this rule correct in sequent calculus? April 17, 2024In Ebbinghaus's book, the following rules occur in sequent calculus: $\dfrac{\Gamma \phi \dfrac{y}{x}}{\Gamma \forall x \phi}$, if y is not free in $\Gamma \forall x \phi$ $\dfrac{\Gamma \phi}{\Gamma \forall x \phi}$, if x is not free in $\Gamma$ I want to know if the following rule is also correct and how to prove its correctness/incorrectness: […]William
- $\exists x\in A \ (P(x))$ versus $\exists x \ (x\in A \land P(x))$ April 17, 2024I have a pretty simple question on mathematical logic. Consider two logical statements $\exists x\in A \ (P(x))$ and $\exists x \ (x\in A \land P(x))$. How are they logically different from each other?RFZ
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