Tag: calculus
Logic on Math StackExchange
- What does the universal quantifier mean? March 27, 2025What does the universal quantifier mean? What does "For any x, F" mean, where "x" is a variable and "F" is a formula? I understand that there are thousands of resources online telling you the meaning of a universal quantifier, but not even Alfred Tarski's explanation makes it clear to me. In my view, "For […]logiclearner
- According to Gödel “this statement is not provable” is paradoxical and the principal problem of logic, is this a way out? [closed] March 26, 2025According to Gödel , "this statement is not provable" is an intensional paradox and the principal problem of mathematical logic. He went even so far to say that the solution to the intensional paradoxes would prove that the human mind is not a machine. I will not use Ackermann's device for brievity sake (1) (1) […]Meta Logician
- Consequence operators can be seen as monads? March 26, 2025Monads on posets are closure operators. The functor $T$ will be a monotone function in the poset $P$. The unit $\eta:1 \to T$ will state that $x \leq Tx$. And the natural transformation $\mu: T^2 \to T$ will state that $T(T(x)) = T(x)$. The monotone condition is given by the monotonicity of $T$: $x \leq […]Lost definition
- Relative strength of metatheory March 25, 2025I have two related questions on the relative strength of metatheory in relation to object theories, especially in the context of formal logic and computer science. a) Metatheoretical strength: It seems there's little study on how to quantify the strength of a metatheory needed to reason about an object theory. For instance, in Morse-Kelley set […]eerio
- How did the requirement of omega-consistency in Godel's incompleteness proof get relaxed? March 25, 2025In Godel's 1931 paper "On Formally Undecidable Propositions..." he introduces and uses the concept of $\omega$-consistency, which is stronger that consistency. I have read somewhere that later on, Skolem was able to relax that requirement, so that any system which is effective (finite set of axioms and schemas), contained arithmetic, and was consistent must have […]Mark Fischler
- Continuum Hypothesis in specific models of $ZFC$ March 25, 2025I know that we cannot prove the consistency of $ZFC$ from within $ZFC$, but for today I will assume that it is consistent, so that it has models. Now, we know that the Continuum Hypothesis ($CH$) is independent from $ZFC$, meaning that we cannot prove $CH$ using the axioms of $ZFC$. Consequently, $ZFC$ must have […]Earnur
- Duality theory - proving that functor has an adjoint March 24, 2025What are some ways to prove that a functor has an adjoint? I have defined algebras for my logic which extends modal logic by adding an additional operation and would like to check if this logic has an appropriate duality theory. Usually, this takes the form of adjunction of general frames and algebras for modal […]tses
- What is a word? March 24, 2025In Classical Mathematical Logic: The Semantic Foundations of Logic by R.L. Epstein, he says the following: When we reason together, we assume that words will continue to be used in the same way. That assumption is so embedded in our use of language that it’s hard to think of a word except as a type, […]R004
- Which sentences can we work with in Classical Mathematical Logic? [duplicate] March 24, 2025In Classical Mathematical Logic: The Semantic Foundations of Logic by R.L. Epstein, he says the following: Which declarative sentences are true or false, that is, have a truth-value? It is sufficient for our purposes in logic to ask whether we can agree that a particular sentence, or class of sentences as in a formal language, […]R004
- Missing Multiplier March 23, 2025For this plumbing job, you are tasked to create a water pipe connection which MULTIPLIES the water volume. There are 3 pipes pumping water into the site: One pipe is connected to the city water supply, transporting effectively infinite volume of water. The two other pipes have a limited water volume. (X and Y) In […]fiqcerzvgm
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