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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Are "equi-expressivity" relations always congruences on Post's lattice?
Given a clone $\mathcal{C}$ over the set $\{\top,\perp\}$, let $\mathsf{FOL}^\mathcal{C}$ be the version of first-order logic with (symbols corresponding to) elements of $\mathcal{C}$ replacing the ...
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Two equivalent statements about formulas projected onto an Ultrafilter
Question 1:
In the same language, let $ X $ be a nonempty set, and let $ \{ (\forall x_{x(i)} f(i)) \ | \ i \in X \} $ be a set of formulas. We use $ x(i) $ to denote the index of the variable on ...
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Whence compactness of automorphism quantifiers?
The following question arose while trying to read Shelah's papers Models with second-order properties I-V. For simplicity, I'm assuming a much stronger theory here than Shelah: throughout, we work in $...
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Is there a substructure-preservation result for FOL in finite model theory?
It's well-known$^*$ that the Los-Tarski theorem ("Every substructure-preserved sentence is equivalent to a $\forall^*$-sentence") fails for $\mathsf{FOL}$ in the finite setting: we can find ...
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Inevitable consistency strength, take 2
This is an attempt to repair this old question of mine.
Say that a good candidate is a first-order sentence $\varphi$ in a language $\{<,...\}$ containing a distinguished binary relation symbol $&...
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Which sentences are "strategically preserved"?
Below, everything is first-order.
Say that a sentence $\varphi$ is strategically preserved iff player 2 has a winning strategy in the following game:
Players 1 and 2 alternately build a sequence of ...
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Is there a computable model of HoTT?
Among the various models of homotopy type theory (simplicial sets, cubical sets, etc.), is there a computable one?
Can the negative follow from the Gödel-Rosser incompleteness theorem?
If there is no ...
user507115
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Book recommendation introduction to model theory
Next semester I will be teaching model theory to master students. The course is designed to be "soft", with no ambition of getting to the very hardcore stuff. Currently, this is the syllabus....
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Is there an abstract logic satisfying the Löwenhein-Skolem property for single sentences but not for countable sets of sentences?
An abstract logic satisfies the LS property for single sentences if each satisfiable sentence has a countable model. Similarly, the LS property for countable sets of sentences holds if every ...
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What is the theory of computably saturated models of ZFC with an *externally well-founded* predicate?
For any model of $M$ of ZFC, we can extend it to a model $M_{ew}$ with an "externally well-founded" predicate $ew$. For $x \in M$, We say that $M_{ew} \vDash ew(x)$ when there is no infinite ...
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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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