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Logic on Math StackExchange
- Reference Request: A general approach to forcing, and some clarification December 4, 2024I've wanted to understand forcing for quite some time now, but I wish to understand it in the general sense of formal logic, rather than just the specific case of set theory, which I find quite confusing anyways. However, I'm struggling to find sources which take this approach, making me doubt it exists. I recall […]Joseph_Kopp
- Prove ¬A ⊢ (A → B) [closed] December 4, 2024Prove ¬A ⊢ (A ⊃ B) using this axioms and modus ponens: A⊃(B⊃A) (A⊃B)⊃((A⊃(B⊃C))⊃(A⊃C)) A⊃(B⊃A∧B) A∧B⊃A A∧B⊃B A⊃A∨B B⊃A∨B (A⊃C)⊃((B⊃C)⊃(A∨B⊃C)) (A⊃B)⊃((A⊃¬B)⊃¬A) ¬¬A⊃AArthur
- Any way to show that second-order logic is incomplete by means other than using Gödel's incompleteness theorem? December 4, 2024I know a way to show second-order logic is incomplete (in the sense of full semantics). the proof is as follows. the second-order arithmetic $\mathbf{Z}_2$ or the second-order peano arithmetic $\mathbf{PA}_2$ are consistent r.e. theories and the Robinson arithmetic $\mathbf{Q}$ can be trivially translated into these theories, so there is an arithmetic formula $G$ for […]Rosser
- (p, q) ∈ S only if p ∧ q is a tautology. Assessing the relation S. December 4, 2024This is the first time I come across a question that combines relations and logic. It is said that: “Let $𝑆$ be a relation, which is defined on the set of propositional formulas as follows $$ \left(𝑝,𝑞\right) \in 𝑆\quad {\bf iff}\quad 𝑝 \wedge 𝑞\ \mbox{is a}\ tautology”. $$ I need to assess if $S$ is […]matinaros
- Axiomatizability of inclusion over any powerset December 4, 2024Is class $(\mathcal{P}(A),\subseteq)$ for arbitrary $A$ axiomatizable? One of the assignments I got was to prove that it is not, but a friend of mine suggested this structure is a complete atomic Boolean Algebra (strictly speaking isomorphic to one) and thus axiomatazible since Boolean Algebras are axiomatizable. So my question is whether his reasoning holds […]Rikimaru
- Best tool for proving that a (big) first order model is actually the model of some (big) first order theory. December 4, 2024I have to check that a (big: several hundreds axioms) first order logic theory is satisfiable. There are various authomated provers for first order logic (e.g. Vampire) that could tell me if the theory is satisfiable or not. However, they all fail (they never terminate or run out of resources), due to the size of […]kataph
- $x\lt Sy\iff x\lt y\lor x=y$ can be deduced from Robinson arithmetic $Q$? December 4, 2024We can define new predicate symbol "$\lt$" in Robinson arithmetic $Q$ by adding the axiom $x\lt y\iff\exists z (x+Sz=y)$.With this definition, $Q$ can prove that (1) $\lnot(x\lt0)$ because for all $x$, $x+Sz=S(x+z)\neq 0$ by the axioms Q1 and Q5. And we can also prove in $Q$ that (2) $x\lt Sy\iff x\lt y\lor x=y$. It is […]smooth manifold
- Question About Axiom Group 5 in Enderton's Logic Textbook December 3, 2024In Enderton's Textbook "Introduction to Mathematical Logic", he specifies 6 Axiom Groups on Page 112 Chapter 2.4 which are used in conjunction with their generalizations, a set of hypothesis $\Gamma$ and Modus Ponens to form deductions in first order logic. The Axiom Group I am confused about is Axiom Group 5, i.e for any variable […]NiraDoesMath
- Number of countable models of DLOWE with an increasing order-isomorphism. December 3, 2024Recently I tried to answer this question by taking the DLOWE and an ascending sequence of constants ($c_1 < c_2 < c_3 \cdots$) construction and encoding it in a finite language with a finite axiomatization. If we think of our DLOWE as $(\mathbb{Q}, \le)$, then the sequence of constants converges to a rational, converges to […]Greg Nisbet
- Reference Request: Second Order ZFC December 3, 2024I've seen a fair bit of discussion about second order ZFC (Second order ZFC, intuition required, How is second-order ZFC defined?, A question about Second-Order ZF and the Axiom of Choice, A question about Second-Order ZFC and the Continuum Hypothesis) but I can't find anything about it in my books. Are there good books which […]daRoyalCacti
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