Tag: perfect circle
Logic on Math StackExchange
- Does truth not exist in ZFC, or is it merely not definable? November 16, 2025I am having a lot of trouble with the concept of Tarski's undefinability theorem as it relates to set theory. Tarski's undefinability theorem says that there is no formula $Tr$ on the natural numbers such that $Tr(\ulcorner A \urcorner) \leftrightarrow A$ So, as far as I understand it, the undefinability theorem says that a truth […]William Oliver
- Syntactic consequence implies semantic consequence [duplicate] November 16, 2025To give a slight background, I am not familiar with any theory of modern logic. I'd been reading Greenberg's book on Geometry where he claims the following. Suppose we have a statement in the formal system but don't yet know whether it is a theorem, i.e., we don't yet know whether it can be proved. […]Jay
- What counts as an argument, technically speaking? November 15, 2025A := It rains. B := I should pick an unbrella. It rains, therefore I should pick an umbrella. p1) A c) B Consider the above premise and conclusion. Initially I'd thought that it's an invalid deductive argument, but now I am not sure whether it is even an argument at all. Is there a […]Loki
- Is there a complete proof system for deriving some axiom schema from others? November 14, 2025Say that an axiom schema is an algorithm $A$ that produces a family of first-order statements (I believe we can recursively enumerate all the algorithms that produce well-formed sentences). Given axiom schemas $A_1$ and $A_2$, say that $A_1\models A_2$ if any model $\mathcal{M}$ of all the statements enumerated by $A_1$ is also a model of […]Pineapple Fish
- What is Gödel's argument for why his proof for a single system applies to all systems November 13, 2025I'm having trouble understanding how Gödel extrapolates from a consistent formal system to any formal system. For reference, his First Incompleteness Theorem states: Any consistent formal system $F$ within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of $F$ which can neither be […]Ben
- Ordinal definability of a minimal witness in L(R, Hom) November 13, 2025Recall that for a transitive set $X$, $L(X)$ is class of sets constructible from $X\cup\{X\}$. For some reason I need to show that every element of $L_\alpha(X)$ is ordinal definable over $L_\alpha(X)$ using parameters from $X$ [1]. I am proving this by induction on $\alpha$, but given limit $\alpha$, if $x\in L_\beta(X)$, then we only […]Akira Satou
- 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
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