Tag: visual math
Logic on Math StackExchange
- Cardinality of the set of measure zero sets June 7, 2023I have been thinking about this question for a while now and found nothing on the matter so far. Assuming the continuum hypothesis (or maybe also for the case that we assume that it is false), what is the cardinality of the set of measure zero subsets of the reals. Obviously it is more than […]Staub und Dreck
- Cardinality of the measure zero sets [closed] June 7, 2023Assuming the continuum hypothesis, what is the cardinality of the set of all subsets of the reals of measure zero?Staub und Dreck
- Is the intersection of two definable sets also definable? June 7, 2023"K₁ and K₂ are definable sets, prove K₁∩K₂ is also a definable set." How do I prove it? I managed to prove for the case of uniting both sets but idk how to do it in the case of intersection. Thanks.Future Hendrixxx
- Is this a free ultrafilter June 7, 2023Suppose $U$ is a free ultrafilter on $\mathbb{N}$. I'm trying to understand what is the ultrafilter generated by the collection $$G=\{E\subseteq\mathbb{N}: E=A+1, \ for \ some \ A\in U\}$$. I think this subset itself is not even a filter (for every set, the minimum element is at least 2, and so $\mathbb{N}$ can't be included). […]user124910
- What are the names that you should give to statements which can be true or false? June 6, 2023I'm compiling a set of notes for high school students about logic and proof and at one point, I write the following: Let $A$ and $B$ be statements. Then $A \implies B$ means ''if $A$ is true then $B$ is true.'' But the following was my definition of a statement: A (mathematical) statement is a […]James
- Epistemic puzzles and the general necessitation-iteration rule RK June 6, 2023So I came across epistemic puzzles, or specifically the "Surprise Exam"/"Prediction Paradox" and found some explanation in Wesley H. Holliday's paper "Simplifying the Surprise Exam", however there is a normal modal logic K used together with the rule: $$RK_{i}\quad\frac{(\varphi_{1}\wedge\cdots\wedge\varphi_m)\rightarrow \psi}{(\Box_{i}\varphi_{1}\wedge\cdots\wedge\Box_{i}\varphi_m)\rightarrow \Box_{i}\psi}$$ How is this rule semantically and syntactically sound for the logic K? If $m=0$ […]jjbinks
- Need help with using fitch system for this proof: Given ¬q, (¬p⇒(¬q⇒¬r)), (s∨r), (s⇒t), and (p⇒t), prove t. [duplicate] June 6, 2023I am having difficulty using the Fitch proof system for the following proof: Given ¬q, (¬p⇒(¬q⇒¬r)), (s∨r), (s⇒t), and (p⇒t), prove t. I'm able to use the following rules: Reiteration, Negation introduction, Negation elimination, And introduction, And elimination, Or introduction, Or elimination, Assumption, Implication elimination, Biconditional introduction, Biconditional elimination, Universal introduction, Universal elimination, Existential introduction, […]JCKing87
- What is the relationship between ultrafilters and propositional theories? June 6, 2023I am learning more about how to use ultrafilters by using them to prove several of the typical results which appear as applications of propositional (or first-order) compactness. Generally speaking, as I try to work out the proofs, I begin to see that ultrafilters produce precise descriptions/specifications certain desirable points (in the sense of what […]John
- Does uniqueness imply existence for the solution of a finite dimensional linear system? June 6, 2023I have a rather naive question concerning a logical implication. Fact: A finite dimensional linear system has either no solution, a unique solution or infinitely many solutions. Now to my question: Suppose I am given a finite dimensional linear system from which I do not know whether a solution exists but I do know that […]Rhjg
- How would you translate this sentence into first-order logic (bolded part)? June 6, 2023From the Wikipedia page of Laramie County: Laramie County is a county located at the southeast corner of the state of Wyoming. As of the 2020 United States Census, the population was 100,512 or 17.4% of the state's total 2020 population, making it the most populous county in Wyoming, but the least populous county in […]Lisramic
Recent Comments