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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Comparing semiring of formulas and Lindenbaum algebra
This is motivationally related to an earlier question of mine.
Given a first-order theory $T$, let $\widehat{D}(T)$ be the semiring defined as follows:
Elements of $\widehat{D}(T)$ are equivalence ...
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Can we see quantifier elimination by comparing semirings?
This question came up while reading the paper Hales, What is motivic measure?. Broadly speaking, I'm interested in which ideas from motivic measure make sense in arbitrary first-order theories (or ...
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Is there a minimal (least?) countably saturated real-closed field?
I heard from a reputable mathematician that ZFC proves that there is a minimal countably saturated real-closed field. I have several questions about this.
Is there a soft model-theoretic construction ...
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Chevalley's theorem on valuation spectra
In the paper On Valuation Spectra (Section 2, Page 176), Huber and Knebusch asserted that: if the ring map $A\to B$ is finitely presented then the associated map of valuation spectra $\mathrm{Spv}(B)\...
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On "necessary connectives" in a structure
Given a clone $\mathcal{C}$ over $\{\top,\perp\}$, let $\mathsf{FOL}^\mathcal{C}$ be the version of first-order logic with connectives from $\mathcal{C}$ in place of the usual Booleans. Given a clone $...
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Mostowski's absoluteness theorem and proving that theories extending $0^\#$ have incomparable minimal transitive models
This question says that the theory ZFC + $0^\#$ has incomparable minimal transitive models. It proves this as follows (my emphasis):
[F]or every c.e. $T⊢\text{ZFC\P}+0^\#$ having a model $M$ with $On^...
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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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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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