Tag: Wau
Logic on Math StackExchange
- What is it we are actually doing when we assume a statement $A$ for the sake of argument? November 7, 2025A very simple question has been bugging me for a very long time now: what is it we are actually assuming when we assume $A$ during natural deduction proofs? (i.e. write "Suppose $A$...") Or 'What are the actual semantics of supposition?' Here are some possibilities: One possibility is that we mean "Assume $A$ is true..." […]William Oliver
- Do equality axioms have to be stated in the meta language first? November 7, 2025This question may sound silly, but there is something about the semantics in first order languages which confuses me a bit. Im using Ebbinghaus,Flum,Thomas:Einführung in die mathematische Logik, as well as Shoenfield:Mathematical Logic for reference They define (I translated the :gdw to :iff): $$\vDash \mathfrak{I}(x_1 \equiv x_2) \qquad \mathrm{:iff} \qquad \mathfrak{I}(x_1) = \mathfrak{I}(x_2)$$ and for […]Alexander Wagner
- Is there a set which is not made fundamentally made from emptyset via axioms? [closed] November 7, 2025Sorry for the wierd sounding title, I am not a set theorist and question won't be well-posed, and in natural language. I also edited the question to be clearer hopefully. Comments about integers are due to original posting My thinking: In a specific set theory (ZF or ZFC) there is a primordial set $\phi$ which […]Rias Gremory
- What type theory really is? November 7, 2025I just wanted to figure out what type theory is, especially dependent type theory (I'm interested in how they corresponds to locally cartesian closed categories), but I just didn't find any definition of what a type theory is. e.g. in wiki, they say in a type theory you have terms, judgements, rules of inference... but […]Westlifer
- Error or imprecise statement in Mac Lane's "Categories for the Working Mathematician"? November 6, 2025$\DeclareMathOperator{\Nat}{Nat}\DeclareMathOperator{\Map}{Map}\DeclareMathOperator{\Ob}{Ob}\DeclareMathOperator{\Set}{Set}\DeclareMathOperator{\id}{id}$Let the definitions and assumptions in set-theory (i.e. existence of one universe $\mathcal{U}$, classes subsets of $\mathcal{U}$, small sets elements of $\mathcal{U}$) and categories as in Mac Lane's book. Let $D$ be a category with small hom-sets, i.e. a locally small category, and $K:D \to \Set$ be a functor. Then for each object $r$ in […]psl2Z
- are logically equivalent statements required to be compound statements? November 6, 2025If two compound statements are logically equivalent if and only if have the same logical values for all possible combinations of their component statements' logical values, does this imply that logically equivalent statements must be compound? If not, how would one show logically equivalent simple statements?Nate
- Can one construct a liar's sentence using the diagonal lemma and the principle of explosion? November 6, 2025Isn't it possible to form the liar's sentence by using the diagonal lemma to form the sentence "This sentence proves all statements"? I.e. For some classical first order theory $T$ containing $PA$, form the predicate $\psi(x) = \forall s( T, x \vdash s)$ then use the diagonal lemma to produce the sentence $\varphi$ satisfying $\psi(\ulcorner […]William Oliver
- Question about the axiom of substitution in Tao’s Analysis I November 5, 2025I'm quoting a sentence from page 57 of Analysis I by Terence Tao: “We observe that functions obey the axiom of substitution: if $x = x'$, then $f(x) = f(x')$ (why?).” My question is: is it possible to prove the above proposition without using the axiom of substitution? Or at least, is that what the […]user1621760
- Universal function for partial computable functions with a domain closure property [closed] November 5, 2025Let 𝑓 : 𝑁 ⇀ 𝑁 f be a partial computable function such that 𝑛 ∈ D o m ( 𝑓 ) ⇒ 2 𝑛 ∈ D o m ( 𝑓 ) In other words, whenever 𝑓 ( 𝑛 ) f(n) is defined, 𝑓 ( 2 𝑛 ) f(2n) is also defined Question: Does the […]ali jahani
- relative frequencies of logical operators in math [closed] November 3, 2025Does anyone know about how often one sees each logical operator in math relative to the others? The five I learned about in my proofs class were OR, XOR, AND, NOT, and IMPLIES, while biconditional implication is XNOR. Are NOR and NAND (or others I might be missing) ever really used in math at all? […]Nate
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