Tag: OpenLab8
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- Relation between Tarski's conception of truth and (implicit) Axioms November 11, 2025I am trying to understand the original paper of Tarski Concept of Truth in the formalized languages, as printed in his collected works. I have read introductory texts from Shoenfield Mathematical Logic (only first 5 chapters thoroughly) and Ebbinghaus, Flum, Thomas Introduction to mathematical logic, as well as early graduate course in logic. but other […]Alexander Wagner
- What does $\forall x\exists x P(x)$ mean? [duplicate] November 10, 2025Is this even a valid formula syntactically? Implies "for all $x$ there is an $x$" that the $\forall x$ is redundant, because obviously there is an $x$? Or is the variable in the new context (after $\forall x$) an other entity and treated like a differently named variable (let's call it $y$)? So basically is […]Janek
- The necessity of introducing CwF for the categorical semantics of Martin-Löf theories? November 10, 2025This question involves two simple subquestions regarding the intuition of necessity of introducing CwF (Category with families) as categorical semantics of Martin-Löf theories. (Note that I'm not familiar with categorical logic and type theory so some questions may not even make sense.) First of all I want to know what does the coherent problem break, […]Westlifer
- Empty Set question from Terence Tao's Analysys I [closed] November 9, 2025Axiom 3.2 (Empty set). There exists a set ∅, known as the empty set, which contains no elements, i.e., for every object x we have x ∈ ∅. The empty set is also denoted {}. Note that there can only be one empty set; if there were two sets ∅ and ∅` which were both […]g0r
- Guessing the outcome of a coin toss with a probability greater than 0.5 November 9, 2025I stumbled across an answer about The envelope paradox which states that: Let Player 1 write two different numbers on two slips of paper. Then player 2 draws one of the two slips each with probability $\frac{1}{2}$ and looks at its number $x$ and has to guess wether $x$ is the larger or the smaller […]math_survivor
- Arguments vs Material conditionals? [closed] November 8, 2025An argument has premises $p$ and one conclusion $c$. Can I write the argument as $p \to c$, in the form of Material conditional? What are the differences between arguments and material conditionals? More-over , arguments have the property validity, and we can say whether an argument is valid. Does it make sense to say […]Jason Cho
- 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? 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 Note: this formulation based on property $P$, was due to one of the comments. The term "a […]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
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