Written work – Due Tuesday, November 15, in class:
Chapter 10 p167: 1, 2, 5, 10, 15
Chapter 10 p167: 25, 28, 30
WeBWorK – none
OpenLab – none
Project – First draft of your group paper is due in class on Thursday, 11/17.
Logic on Math StackExchange
- Is it possible to define the notion of an indexed family in higher-order logic? May 29, 2023I have been trying to define the notion of a product of second-order classes using (finitary) second-order and if needed third-order logic. It seems to be possible to define the product of finitely many classes, because I can just express this using a finite second-order sentence. The problem is that, when I try to define […]user867078
- How is the logical connective $\bot$ used in formulae May 29, 2023We are working on a question in which we have to prove that a certain set of logical connectives is functionally complete. It is said that we are allowed to use the fact that the set $\{\bot,\to\}$ is functionally complete. Our question is: how is the logical connective $\bot$ used in formulae? Is it possible […]UnrulyTank
- The existence of a sentence which ensures isomorphism May 29, 2023Let $\mathcal{L}=\{P\}$ where $P$ is a unary relation symnbol and let $\mathcal{M}$ a finite $\mathcal{L}$-structure. Show that there exists $\sigma \in Sent_{\mathcal{L}}$ such that for all $\mathcal{L}$-structures $\mathcal{N}$, we have $$\mathcal{N} \vDash \sigma \text{ if and only if } \mathcal{M} \cong \mathcal{N}$$ I am having issues with this problem, I have tried to prove it […]Superdivinidad
- What are all the restrictions on existential generalization (When CAN'T we use it)? May 28, 2023I know that I am using existential generalization wrong here, I don't know exactly how but the conclusion is absurd. Let $F$ be any functional predicate. $F(x)=F(x)$ axiom $\forall x(F(x)=F(x))$ universal generalization $x$/$x$ (1) $\exists y\forall x(F(x)=y)$ existential generalization $y$/$F(x)$ (2) $\forall x(F(x)=z)$ existential instantiation $z$/$y$ (3) $F(a)=z$ universal instantiation $a$/$x$ (4) $F(b)=z$ universal instantiation […]Isaac Sechslingloff
- Satisfiability in an Heyting algebra implies satisfiability in a Boolean algebra for propositional logic? May 28, 2023Let $\mathcal{L}$ be a propositional language and let $\text{Prop}(\mathcal{L})$ be the set of all the propositions of the language $\mathcal{L}$. Let $(H,\wedge,\vee,\rightarrow,1,0)$ be an Heyting algebra. An evaluation of the propositions of $\mathcal{L}$ in $(H,\wedge,\vee,\rightarrow,1,0)$ is a map $V:\text{Prop}(\mathcal{L}) \to H$ such that $V(\top)=1$, $V(\bot)=0$, $V(P \wedge Q)=V(P) \wedge V(Q)$, $V(P \vee Q)=V(P) \vee V(Q)$ […]effezeta
- Are there "Godel encodings" for non-arithmetic theories? May 28, 2023This is probably a very naive question, but is there something about Godel encoding that is essentially arithmetical, or is it possible to construct analogous mappings between the objects studied in a theory and statements in the theory itself for non-arithmetical theories (i.e, theories whose objects of study are not numbers). Are there any notable […]Rando McRandom
- How well should I know understand the theorem before proving it? May 28, 2023Im reading "How to prove it" by velleman, and in a lot of the exercises you need to prove some theorem is true. How well should I understand why the theorem is actually true before proving it? For me it's pretty easy to prove something just by following the rules, but idk if that's smart. […]lightyourassonfire
- Iff propositions where both directions require choice? May 28, 2023Recently, I have been revising a basic course on noncommutative rings and modules over them. One proposition proven early on is all left modules over $R$ are free iff $R$ is a division ring and an interesting point is that, at least in the standard proofs we were given (and I have seen repeated in […]Isky Mathews
- Satisfiability in greater languages May 28, 2023Let $\mathcal{L}, \mathcal{L}'$ two languages in first order logic such that $\mathcal{L}\subset \mathcal{L}'$, $\Gamma \subseteq Form_{\mathcal{L}}$ and $\varphi \in Form_{\mathcal{L}}$. Prove that if $\Gamma \vdash_{\mathcal{L}'} \varphi$ then $\Gamma \vdash_{\mathcal{L}}\varphi$. In my book, there is a proof when $\mathcal{L}'$ is an extension of $\mathcal{L}$ with only constant symbols, but this is the general case. I am […]Superdivinidad
- How to solve $(x-1)(x-2)=0$ constructively? May 28, 2023I want to prove that $$(x-1)(x-2)=0\Leftrightarrow x=1, 2$$ $\Leftarrow$ is easy. The problem is $\Rightarrow$. Assuming $x\neq 1, 2$, we can derive $1=0$ by dividing both sides of $(x-1)(x-2)=0$ by $x-1$ and $x-2$. Thus we get $\lnot \lnot (x=1, 2)$. However, intuitionistic logic cannot eliminate double negation.BonBon
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