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Logic on Math StackExchange
- Making the notion that if a statement is true for all naturals smaller than a very large number then the statement is true for all naturals rigorous [closed] April 29, 2025There are plenty of examples of statements of the form "$P(n)$ is true for every $n\in\mathbb{N}$" such that there is a "very large" $m$ for which $P(n)$ is true for every $0MSEU
- $\bot$ cannot be defined in positive intuitionic logic? [duplicate] April 29, 2025If $\bot$ could be defined in positive intuitionistic logic, then minimal logic would be positive intuitionistic logic: see here for the axioms. Now, how to proof that $\bot$ is not definable in positive intuitionistic logic? My attempt: we can model implication of positive intuitionistic logic on real line $(0,1]$ with usual order. We can be […]Lost definition
- Why isn't 'heterological' autological? April 29, 2025In simple type theory, one can formulate the heterological paradox: Het(x) := ∃φ (x means φ & ¬φx) Let x = ‘Het’, then φ=Het(x), then Het(‘Het’) ¬Het(‘Het’). But in ramified type theory, we can no longer have the paradox since Het(x) is of an order higher than φ(x), so it cannot fall under the […]John Smith
- Designing $6$ yes/no questions to find a number from $1$ to $8$, if one answer is allowed (but not required) to be a lie April 29, 2025I have a logic puzzle where a person thinks of a number between 1 and 8. I am allowed to ask yes/no questions to figure out the number, but the person is allowed to lie exactly once (or possibly not at all). It was shown mathematically that at least 6 yes/no questions are needed. (Because […]BAAAAE
- Is this characterization of paraconsistent logic $C_\omega$ right? April 28, 2025Priest characterization of $C_\omega$ can be found here. It says that: To obtain da Costa’s system $C_\omega$, instead of the positive fragment of classical logic, we start with positive intuitionist logic instead. $C_i$ systems for finite $i$ do not rule out $(A^n \land \land \neg A^n \land A^{n+1})$ from holding in a theory. By going […]Lost definition
- A Weakly 2-Random Set that is both Hyperimmune and Generalized Low April 28, 2025In a 2007 paper, Nies, Montalbán and Lewis build a weakly 2-random set that is not generalized low, hence separating weak 2-randomness from randomness. This is done by constructing a 1-random hyperimmune-free not-generalized-low set, which by properties of randomness implies that this set is weakly 2-random but not 2-random. In the same paper, they show […]Robly18
- Axiomatic system for propositional logic is an empty sequence, why? [closed] April 28, 2025I study physics as my main degree, but I am currently attending lectures on differential geometry. We started with logic and I have a question about propositional logic and axiomatic systems. The definition that we were given about axiomatic systems is: "An axiomatic system is a finite sequence of propositions or propositional schemes $a_1,a_2...a_N$ which […]imbAF
- Is it known whether Free Complete Heyting Algebras Exist? April 27, 2025I was reading the wikipedia page for heyting algebras, and it made the claim that "it is unknown whether free complete heyting algebras exist". It came unsourced, but by tracking the edit I was able to source the claim to page 34~35 of Stone Spaces by Johnstone, where he says something which I can see […]Glubs
- a property between compactness and non-compactness in logic April 27, 2025Background and motivation: Without specifying exactly what "a logic" is (since any one definition would be unnecessarily restrictive for the purposes of this question), we can say that a logic is compact if it satisfies the analogue of the compactness theorem for FOL, that is, if a set of sentences is finitely satisfiable, then the […]Carlyle
- Are $1$, $2$, $3$... the only definable natural numbers? April 27, 2025Of course $1$, $2$, $3$... etc have defining axioms. If we're working with an $\omega$-inconsistent theory we can also have a natural number $m$ satisfying $\neg P(m)$ even though $P(1)$, $P(2)$, $P(3)$... etc are all provable. But we don't have uniqueness, so we can't add a defining axiom. Can there ever be a definable natural […]Pineapple Fish
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