Tag: visual math
Logic on Math StackExchange
- 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
- Is quantum mechanics complete -mathematically - [closed] April 26, 2025I have a question If we consider gödel's incompleteness theorems, then quantum mechanics mathematical formalism isn't coherent nor complete, then why did the Copenhagen interpretation scientists say that the mathematical foundation of quantum mechanics is no subject of objection ? "Nous tenons la mĂ©canique des quanta pour une thĂ©orie complĂšte, dont les hypothĂšses fondamentales, physiques […]Mohammed ramy Cherif
- In Vellemanâs How to Prove It Example 1.1.2 (1), should âEither John went to the store, or weâre out of eggsâ be modelled using exclusive OR? [duplicate] April 26, 2025Iâm working through Daniel Vellemanâs textbook How to Prove It: A Structured Approach (second edition). In Example 1.1.2 (1). The book analyses the statement: Either John went to the store, or weâre out of eggs. It models this as: $$P \lor Q$$ where: P = âJohn went to the storeâ Q = âWeâre out of […]georgeamccarthy
- Does this Weird Correspondence of $\Box$ to $\top \to$ to the S4 Axioms Allow for a Companion to IS4 with No Modal Operators? April 26, 2025This just struck me as weird while I was reading a bunch of papers on modal decision procedures for intuitionistic propositional logic: N : If $A$ is a theorem, derive $\Box A$. ~ If $A$ is a theorem, derive $\top \to A$. K : $\Box (A \to B) \to (\Box A \to \Box B)$ ~ […]Joshua Harwood
- Help walking through difficulties understanding the difference between $\forall x$ and infinite conjunction April 26, 2025I (and possibly others) naively thought of $\forall$ as an infinite conjunction (only vaguely aware of others saying to avoid this interpretation of the $\forall$ sign). Years later I'm confronting this belief. I've read over Derek Elkins' blog post on the matter. I only partially probably have the logical prerequisites to approach the problem, but […]Pineapple Fish
- Does $p,q â S$ require $p$ and $q$ to be distinct? Name binding in mathematics. [closed] April 25, 2025I have noticed that different authors use different rules for such simple statements such as "Let there be two numbers $p$ and $q$." Most allow the possibility that $p$ and $q$ are the same number. Others treat $p$ and $q$ as necessarily distinct; for example, Stephenson in An Introduction to Matrices, Sets and Groups specifically […]Steven Thomas Hatton
- Ordering the set of all filters on a partial order. April 25, 2025Fix two partial orders, $R$ and $R'$, on the same set $X$. Denote by $F(R)$ and $F(R')$ the set of all filters of $R$ and $R'$ respectively. Specifically, for $f \subseteq X$ we have $f \in F(R)$ if $f$ is non-empty $x \in f$ and $xRy$ implies $y \in f$ $x,y \in f$ implies there […]201p
- Explanation of a fallacy during Structural Induction [closed] April 25, 2025Here is the description of Structural Induction extracted from "Discrete Mathematics and Its Applications - 8th edition" by Kenneth Rosen: BASIS STEP: Show that the result holds for all elements specified in the basis step of the recursive definition to be in the set. RECURSIVE STEP: Show that if the statement is true for each […]Vlad Mikheenko
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