# Tag: perfect circle

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- What are the proof-theoretic strengths of Ramsey's theorems? March 31, 2023Consider the following statements: Finite Ramsey Theorem ($\mathrm{FRT}$): For any $r,k,p\in\mathbb{N}$, there exists $N\in\mathbb{N}$ such that for any set $S$ of cardinality $N$ and any function $c:[S]^{p}\rightarrow [r]$ there exists a subset $H\subseteq S$ of cardinality $k$ such that $c|_{H}$ is constant. By $[r]$ we mean the set $\{1,\ldots r\}$, and the function $c$ is […]John
- Truth sets in mathematics March 31, 2023The definition of mathematical truth is confusing. Let P and Q denote the statements 'I'm a man' and '$1Nagbu
- Are Z and ZC equiconsistent? [closed] March 31, 2023Let Z be Zermelo's system without choice (C) and without axiom of foundation (AF), namely extensionnality, pair, union, power set, separation, empty set, infinity. I have two questions : (1) is Z equiconsistent with Z+C ? (2) is Z+AF equiconsistent with Z+AF+C ? (I know that ZF is equiconsistent with ZF+C)Ivan Marin
- Typed logic vs many-sorted logic March 31, 2023I am confused as to what is the difference between many-sorted logic and typed logic. Are they the same thing? If not, what are the differences?BonBon
- Textbook of logic and sets based on type theory [closed] March 31, 2023Are there any textbooks on logic and sets at the level of prior knowledge to calculus that are based on type theory?BonBon
- Q3 assignment week11 Provide an interpretation that makes all of the sentences in the problem true. March 30, 2023Provide an interpretation that makes all of the sentences in the problem true. F(a), F(b), F(c), ∀x∀y(G(x,y) ↔ G(x,x)), ∀x(F(x) → G(x,x)) My answer is: Domain:0,1,2 F(_):0,1,2 G(,):(0,0),(1,1),(2,2),(0,1),(0,2),(1,2),(2,1) a:0 b:1 c:2 Carnap says: Not all formulas are true in this model. Take another look at: ∀x∀y(G(x,y) ↔ G(x,x)). Because G(x,y) G(x,x) , so I write […]user21286568
- Constructive proof of compactness theorem for countable propositional languages March 30, 2023Let $\mathcal{L}$ be a countable propositional language and let $\Gamma$ be a set of propositions of $\mathcal{L}$ (i.e. $\Gamma \subseteq \text{Prop}(\mathcal{L})$). Definition: $\Gamma$ is satisfiable if there exists an evaluation function $V:\text{Prop}(\mathcal{L}) \to \{\bot,\top\}$ such that $V(C)=\top$ for every $C \in \Gamma$ ($\{\bot,\top\}$ is the 2-elements boolean algebra). Compactness theorem for countable languages: if every […]effezeta
- Does Los's theorem hold for the filter $\{I\}$? March 30, 2023We know that $\{I\}$ is a filter over $I$. I'd like to show that $Th(M^I/ F) = Th(M)$ as a consequence of Los's theorem. Now, Los's theorem does not work in general for filters. See: What can we say if we have a filter instead of an ultrafilter in Los's theorem? Does it work for […]user1868607
- How can one show that PA is equal to this? March 30, 2023In some class note, I have seen the alternative definition of PA which is given as following: $$PA=\cup I \Sigma_n=\cup I\Pi_n$$ , while $I \Phi $ is Q equipped with the $\Phi$ induction axioms, given some class of formulas, $\Phi,$ given as below: $$\bigl\{ A(0) \implies (\forall x)(A(S(x)) \implies (\forall x)A(x): A \in \Phi \bigl\} […]io13331
- What is the justfication for splitting up statements and quantifiers? March 30, 2023To explain the title, in proving $X \times Y = \emptyset \iff X = \emptyset \vee Y = \emptyset$, we have the following \begin{align*} X \times Y = \emptyset &\iff \forall x \forall y \left((x,y) \notin X \times Y\right)\\ &\iff \forall x \forall y \left(x \notin X \vee y \notin Y\right) \\ &\iff \forall x […]jacob

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