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Logic on Math StackExchange
- What is "coherence" in mathematical logic? July 16, 2026Section 2.5 (pg. 5) of this paper talks about "coherent" theories. I'm unfamiliar with what seems to be a formal notion of "coherence" in mathematical logic being referenced there. So, I'm wondering if someone can provide a brief description of this formal notion of "coherence" (or reference to a good description). Of course, the word […]NikS
- are there K-saturated structures that are not elementary equivalent? July 15, 2026I am working in the context of Hrushovski constructions for my thesis, using mainly a paper by Baldwin and Shi as a reference. (For transparency this part of the question is copied from Is every $\mathcal{K}$-saturated structure $\mathcal{K}$-homogeneous?) Let $K$ be an hereditary (it is not necessary in the paper, but in the concrete examples […]Tarallino_Birichino
- Algebraic characterisation of $\exists\forall$ and $\exists\forall\exists$ theories? [duplicate] July 14, 2026There are well known criteria for quantifier bounds like this. A (first order, consistent) theory $T$ is $\exists$-axiomatizable iff the class of its models Mod$(T)$ is closed under superstructures, it is $\forall$-axiomatizable iff Mod$(T)$ is closed under substructures, and it is $\forall\exists$-axiomatizable iff Mod$(T)$ is closed under ascending chains. Are there similar characterizations for other […]Susana Santoyo
- Questions on using an ultrafilter to "collapse" a boolean valued model and show that CH is independent of ZFC July 12, 2026I just finished going through a proof that CH is independent of ZFC using a boolean valued model. Here's the proof: Let $P$ be the poset of functions from finite subsets of $\aleph_2 \times \aleph_0 \rightarrow \{0, 1\}$ ordered by reverse inclusion. Let $B$ be its completion. For $a \in \aleph_2, n \in \mathbb{N}, i […]David Lui
- What exactly do we obtain from a concrete formal proof? July 11, 2026Suppose I find a formal proof of a sentence $\varphi$ in PA. What exactly have I obtained from this? Can I say that the mathematical statement expressed by $\varphi$ has actually been proved? At first sight, it seems that all I have shown is a syntactic fact about a formal system: there is a finite […]mathlogic2025
- Is it possible to define a total order on arbitrary arithmetic expressions of a single real variable that respects eventual sign? July 10, 2026By arithmetic expression I mean one consisting of arbitrary combinations of natural number constants, a single real variable x (which may appear any number of times in the expression), and the five arithmetic operators - addition, subtraction, multiplication, division, and exponentiation. I would like to define a total order on such expressions $f$, $g$ such […]stackshifter
- Is there a construction where the universal quantifier is more natural than the existential quantifier? July 9, 2026Universal and Existential quantifiers are dual, so it is usually said that when defining a first-order language, one can take either (or both, of course) one as a primitive. But when taking the categorical/topos theoretical perspective, only the existential quantifier seems to admit a natural construction. Concretely, $\exists_f$ can be defined as a composition with […]prime235711
- What is Tarski's Thesis Saying about logical operators? July 8, 2026Following Gila Sher's treatment of Tarski's Thesis. She defines a structure as tuples $$\langle A,\alpha_1,\ldots\rangle$$ where $A$ is a set (presumably in ZF of some kind, Sher hints at Principia as foundation or "similar theory") with each $\alpha_i$ a "set-theoretic constructs of elements of $A$". Then two structures are isomorphic, denoted $$\langle A,\alpha_1,\ldots\rangle\cong \langle B,\beta_1,\ldots\rangle$$ […]Algeboy
- Exist a unique homomorphism from term algebra to a model? [closed] July 7, 2026Exist a domain $\mathbb{K}$ $\\\mathbb{Q}⊆\mathbb{K}⫋\mathbb{C} \\ ∀a,b∈\mathbb{K}， a^b∈\mathbb{K} ~(if~a=0, ~Re(b)>0)$ Note that $a^b$ is main-branch-choosing exponential function and isn't defined when $a=0 ~Re(b)≤0$. But it doesn't influence saying $a^b$ is closed in $\mathbb{K}$ Because consider {$\mathbb{C},+,\times ,\mathbb{0},\mathbb{1}$}the theory of algebraically closed fields, which is a countable language and has countable elementary submodels. Therefore, we can […]chinesemathnoob
- Reducing adjacent quantifiers of the same type in Peano Arithmetic July 6, 2026I'm currently reading through Reverse Mathematics by John Stillwell. In section 2.8, while defining what it means for a formula in Peano Arithmetic to be $\Pi_m^0$ or $\Sigma_m^0$, he writes that we can "reduce adjacent quantifiers of the same type to a single one," since $$\forall x\forall y\,\varphi(x,y)\iff\forall z\,\varphi(P_1(z),P_2(z))$$ and$$\exists x\exists y\,\varphi(x,y)\iff\exists z\,\varphi(P_1(z),P_2(z))$$ where $P_1$ […]homiamorphism
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