**Week 2 Assignments**

**Written work** – none

**WeBWorK **– Assignment #1 and #2, due Tuesday, September 6th, at midnight.

**OpenLab **– OpenLab #2, due Thursday, September 8th, at the start of class.

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- Does there exist an axiomatic theory that deals with ordered sets of axioms? (As way of formally introducing "conviction" into deductive systems) January 25, 2022Does there exist examples in the literature of a formal system (axiomatic theories) that deals with ordered sets of axioms? The question is philosophically motivated but I believe it lies in the domain of mathematics and logic. All theories that I’ve seen (built on the formalizations establish by logicians in the previous two centuries) consider […]stam_a
- Why this theory is consistent? January 25, 2022I'm not asking how to do these exercises. But I'm very confused about a thing. Question: By hypothesis $T$ is an axiomatization of the class of $L$-structure with an even cardinality. So if $\mathcal{M} $ is a model of $T$ then it has an even cardinality. Also $T_{\infty} $ is an axiomatization of the class […]3m0o
- (A logic question) I think its the right answer, but my explanation could be a bit more precise? January 25, 2022Is (A & B & ~C) a sentence of Sentential Logic? Answer: It is not because there needs to be at least one parentheses out side of two sentences connected by an ampersand. Is this explanation correct?logicmagicianintheair
- Does $N \subseteq M$ implies $L^N \subseteq L^M$? January 25, 2022Let $N \subseteq M$ be two transitive models of $\mathsf{ZFC}$. Let $L^N$ and $L^M$ denote the constructible universes in $N$ and $M$ respectively. Is it true that $L^N \subseteq L^M$? I believe that this is true, which can be proven using Gödel's operations to represent $\operatorname{Def}(L_\alpha)$ for each level $L_\alpha$, but perhaps a much simpler […]Clement Yung
- Define $\{ - \sqrt 2 \}$ using only $\langle =,+, -, \cdot, 0, 1 \rangle $ on $ \mathbb{R} $ [duplicate] January 25, 2022On the structure $\mathbb{R}$ (standard real numbers) and signature $\langle =,+, -, \cdot, 0, 1 \rangle $ define the set $\{-\sqrt 2\}$, i.e. find a formula $\varphi$(a), such that $ \{a \in \mathbb{R}; \varphi(a) \} = \{-\sqrt 2\}$. It is clear to me that I can get $\{\sqrt 2, -\sqrt 2 \}$ easily with $\varphi(a) […]pavelkomin
- Truth tables - understanding if $P$ then $Q$ [duplicate] January 25, 2022I should have understood this much earlier, but for some reason it's just not clicking. When taking an if-then statement, I'm confused about the boolean logic, specifically when $P$ is false. Taking the statement (assuming that $x$ is only positive real numbers) "if $x = 1,$ then $x^2 = 1.$" Clearly, if $x$ is not […]Daanyal Ali Akhtar
- What does close mean in first order logic? January 24, 2022I just ran across this statement in a logic book A predicate can’t be true or false until a specific value is substituted for the variables, and the quantifiers ∀ and ∃ “close” over a predicate to give a statement which can be either true or false. I think I understand the "specific value" part; […]147pm
- Induction to Define Permutation (Propositional Logic)? January 24, 2022For a well-formed formula φ, use induction to define permutation(φ), which is the number of logically equivalent formulas obtained from φ by changing the order of the operands in the logical connectives in φ e.g., permutation(p ∨ q) = 2 because p ∨ q ≡ q ∨ p, permutation(p → q) = 1 because p […]uwu
- Axiomatizability of class of the language $L=\{ = \} $ January 24, 2022I was solving a mock exam and there is a "tricky" question, for me at least. It seem to me impossible that question 1) and 4) are both true. I would love to understand Let $L$ a language with the symbol $ \{=\} $ and $C$ a class of $L$-structure. We say that $C$ is […]3m0o
- Explain why $\varphi$ is a tautology, and $\psi$ is a contradiction (unsatisfiable formula) January 24, 2022Let $\varphi \rightarrow \psi$ be a contradiction with well-formed formulas $\varphi$ and $\psi$. Explain why $\varphi$ is a tautology and $\psi$ is a contradiction (unsatisfiable formula). We have begun propositional logic in class and this is an example for lecture. I am confused how to prove this seeing as it doesn't seem plausible to use […]uwu

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