Questions tagged [model-theory]
Model theory is the branch of mathematical logic which deals with the connection between a formal language and its interpretations, or models.
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Effortless automated proofs for "simple" formulae?
From small cases to all of them. This is in the spirit of 15 theorem see https://en.wikipedia.org/wiki/15_and_290_theorems
EXAMPLE : Suppose you have the following problem: P(a)
For any fixed non ...
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Is Presburger arithmetic in stronger logics still complete?
Originally asked at MSE:
Let $\Sigma=\{+,<,0,1\}$ be the usual language of Presburger arithmetic. Given a "reasonable" logic $\mathcal{L}$, let $\mathbb{Pres}(\mathcal{L})$ be the $\...
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Comparing two fragments of SOL with the downward Lowenheim-Skolem property
For $S$ a set of (parameter-free) second-order formulas and $\mathfrak{A},\mathfrak{B}$ structures, write $\mathfrak{A}\trianglelefteq^S\mathfrak{B}$ iff $\mathfrak{A}$ is a substructure of $\mathfrak{...
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Can we have external automorphisms over intersectional models?
Is the following inconsistent:
By "intersectional" set I mean a set having the intersectional set of every nonempty subset of it, being an element of it.
$\forall S \subset M: S\neq \...
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In finite model theory, is "invariant FOL with $\varepsilon$-operator" unavoidably second-order?
Throughout, all structures are finite.
Say that a class of finite structures $\mathbb{K}$ is $\mathsf{FOL}_\varepsilon^\text{inv}$-elementary iff it is the class of finite models of a sentence in the ...
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Natural Numbers
Let $L$ be a countable language and $M$ be a model of $L^N$ (the realization of $L$ in the natural numbers $N$) in which every recursive unary relation is expressible. Show that $M$ is not recursive.
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Are some interesting mathematical statements minimal?
Gödel's set $\mathrm{L}$, of constructible sets, decides many interesting mathematical statements, as the Continuum hypothesis and the Axiom of Choice.
Are some interesting mathematical questions, ...
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The "first-order theory of the second-order theory of $\mathrm{ZFC}$"
$\newcommand\ZFC{\mathrm{ZFC}}\DeclareMathOperator\Con{Con}$It is often interesting to look at the theory of all first-order statements that are true in some second-order theory, giving us things like ...
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Expressiveness in arithmetic
Let $\mathcal{S}$ be a formal system for arithmetic (e.g. $P$ or $PA$), $f:N^q\rightarrow N^p$ a function of $N^q$ on $N^ p$ and $\alpha(\mathbf{x})$ a formula of $\mathcal{S}$ with $p$ free variables....
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Is $(\mathbb R_{\mathrm{an}}, (x\mapsto x^r)_{r\in\mathbb R})$ still the largest known polynomially bounded o-minimal structure so far?
From Chris Miller's paper in 1995, the structure $(\mathbb R_{\mathrm{an}}, (x\mapsto x^r)_{r\in\mathbb R})$,is the largest known polynomially bounded o-minimal structure as of that time. I wonder if ...
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What is this quotient of the free product?
Previously asked at MSE. The construction here can generalize to arbitrary algebras (in the sense of universal algebra) in the same signature with the only needed tweak being the replacement of "...
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Are the real numbers isomorphic to a nontrivial ultraproduct of fields?
Let $K_1, K_2, \dots$ be a countable sequence of fields, and let $\prod_{\mathcal F} K_i$ be the ultraproduct with respect to some nonprincipal ultrafilter $\mathcal F$.
Question: Can there be a field ...
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Finitely presentable groups are residually finite if and only if they are universally pseudofinite
Suppose $G$ is finitely presentable. Then residual finiteness of $G$ is equivalent to $G$ satisfying the universal theory of finite groups (equivalently, to every existential statement true in $G$ ...
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Do maximal compact logics exist?
By "logic" I mean regular logic in the sense of abstract model theory (see e.g. the last section of Ebbinghaus/Flum/Thomas' book). My question is simple:
Is there a logic $\mathcal{L}$ ...
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Gödel coding and the function $z(x)$
The function $z(x)$ that associates to each formula $\alpha$ of $P$ its Gödel number $z(\alpha)$ is external to the system. How then can expressions in which $z(x)$ be involved be expressed in $P$? ...
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