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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Compactness of existential second order logic and definability of certain quantifiers
It is known (as a slogan) that the "existential fragment of second-order logic (ESO) is compact".
My first question is:
(1) Is ESO compact for:
(a) uncountable languages
(b) languages with ...
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A (second-order) axiomatic characterization of the integers which rules out surreal/hyperreal versions
I've seen it stated, for example here, that the integers are the unique commutative ordered ring with identity whose positive elements are well-ordered.
I understand why the integers are the smallest ...
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the structure on the value group sort of a C-minimal field
Let $K$ be an algebraically closed valued field which is $C$-minimal, as defined, for example, in this article. Examples include pure algebraically closed valued fields, as well as Lipschitz and ...
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Why do we need a transitive model in forcing arguments?
One major approach to the theory of forcing is to assume that ZFC has a countable transitive model $M \in V$ (where $V$ is the "real" universe). In this approach, one takes a poset $\mathbb{P} \in M$, ...
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Constructive Resolution of Toric Singularities via Model Theory
Do there exists some language $\mathcal{L}$ of rational polyhedral cones in rational vector spaces and a theory $T$ over $\mathcal{L}$ whose models $\mathcal{M}$ are resolutions of toric singularities?...
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Consistent sentences with no arithmetically definable models
I've seen a construction of a sentence of first order logic that is consistent, but has no models with underlying set $\mathbb{N}$ and recursive functions and relations. Do there also exist consistent ...
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A kind of saturation property related to forcing notions
Forcing is typically done over well-founded models. There are lots of good reasons for this, but it can feel confining at times. Fortunately, we can equally well force over non-well-founded models! It ...
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Construction of model of arithmetic from an arbitrary model
Let $M$ be a non-standard model of $PA$, $a\in |M|$ be an arbitrary non-standard number and $T$ be a theory of arithmetic. We want to choose a subset $M'\subsetneq M$ such that:
$M'\models PA^-$ (or $...
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Is there a model of ZF+ACC where transfer fails for the definable hyperreals?
In 2003 Kanovei and Shelah constructed a definable hyperreal field. The ultrapower used exploits a fairly large index set so that it is clear that the usual proof of Los and transfer does not go ...
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Model existence and consistency conditions for $\Pi_1^0$ oracles
Let a $\Pi_1^0$ sentence be a sentence asserting that some given Turing machine never halts at the empty input tape. Let Q1 be a (potentially consistently lying) oracle for deciding $\Pi_1^0$ ...
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O-minimal spectrum is a spectral space
I'm trying to understand a proof on "Sheaves of Continuous Definable Functions" (Pillay, Anand. "Sheaves of continuous definable functions." The Journal of symbolic logic 53.04 (1988): 1165-1169.)
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Existentially closed partial orders
Existentially closed linear orders are dense linear orders without endpoints, which are finitely axiomatizable, and occur as order-types of natural mathematical structures such as the rationals or ...
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Example of $\aleph_1$-categorical linear order
Is it possible to have an $L_{\omega_1,\omega}$-sentence $\phi$ in a vocabulary that includes $<$ that satisfies the following?
$<$ is a linear order on a definable subset;
$\phi$ is $\aleph_1$-...
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Is there a Fraisse limit whose automorphism group contains dense but not generic automorphisms?
It is well known that $\mathsf{Aut}(\mathbb{Q},<)$ has generic automorphisms (i.e., a comeagre conjugacy class under the diagonal action) but does not admit ample generics. The automorphism group $\...
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When are generic models not too wild?
This is a question related to ideas raised in http://arxiv.org/abs/1410.1224 and http://arxiv.org/pdf/1405.7456.pdf. Basically, the idea is the following:
Suppose I have a first-order theory $T$. ...
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Are Braid Groups with Finitely many Generators NIP?
I am curious what braid groups (strings in $\mathbb{R}^3$) are NIP. Consider the following:
Let $B_\mathbb{N}$ be braid group with "braids" indexed by the natural numbers (alternatively, the ...
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Interpreting peano arithmetic without parameters
I will accept an answer in the form of references to the literature about my question as well as any other information. I am quite ignorant of the area and that will be clear from my question.
I ...
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Limits of definable maps
For sequences of semialgebraic maps there is the following result:
Let $(f_{n}: ]0,1[^d \to ]0,1[)_{n \in \mathbb{N}}$ be a sequence of continuous semialgebraic maps of bounded degree such that $(...
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Spectrum Problem for Higher-Order Logic
Definitions. Given a sentence $\varphi$ of $n$th-order logic, we define the spectrum of $\varphi$ to be the set of cardinalities of finite structures that satisfy $\varphi$. A set $X\subseteq\mathbb ...
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Expressive power of $\omega$-order logic
According to the article Second-order and Higher-order Logic
from the Stanford Encyclopedia of Philosophy,
there is no need to stop at second-order logic; one can keep going. [...] we can allow ...
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$\mathcal A\equiv\mathcal B\implies \mathcal A\cong\mathcal B$ for finite $\mathcal L$-structures where $\mathcal L$ is an infinite signature
Let $\mathcal L$ be an infinite signature and $\mathcal A$, $\mathcal B$ two finite $\mathcal L$-structures such that
for each first-order $\mathcal L$-sentence $\varphi$, $$\mathcal A\models\varphi\...
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Definable curves in definable sets
Suppose that I have an unbounded subset $X \subset \mathbb{R}^n$, definable in the $o$-minimal structure $\mathbb{R}_{an, exp}$. Is it possible to find an unbounded, analytic and definable curve (i.e. ...
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Can one satisfaction class code another?
Let $M$ be a model of ${\sf ZFC}$. A satisfaction class $S$ for $M$ is subset of $M$'s ordered pairs which satisfies in $M$ the standard Tarskian compositional axioms. E.g.:
$M\vDash \forall \phi, \...
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Quantifier elimination of pp-subgroups of modules
This is a model-theoretic questions. Let $M$ be a $R$-module. Our language will be the standard language of modules, i.e. the language of abelian groups together with an unary function symbol for ...
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Least inner model of ZF without power set axiom
I'm interested in the existence and properties of an analogue version of $L$ for models of ZF$^-$ (ZF without the power set axiom), which for simplicity I'll call $L^-$. By "analogue" I mean the least ...
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Does the category of (algebraically closed) fields of characteristic $p$ change when $p$ changes?
Let $\mathrm{ACF}_p$ denote the category of algebraically closed fields of characteristic $p$, with all homomorphisms as morphisms. The question is: when is there an equivalence of categories between $...
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What is the Cantor-Bendixson rank of the space of first order theories?
Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its ...
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A question concerning model theory of groups
Several days ago, Professor Martin Bridson gave a very nice talk in my department. Some questions concerning his talk came into my brain Since I am neither a model theorist nor a algebraist, I am not ...
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Independence in mathematics
While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over ...
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Godel's proof of Completeness
Where could I find a detailed exposition in English of Godel's proof (not Henkin's) of Completeness Theorem for first order logic? The wikipedia article omits certain details that I am not clear about,...
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Uniform elimination of imaginaries
Does the following principle follow from uniform elimination of imaginaries?
For every formula $\varphi(x;y)$ there is a formula $\vartheta(x;z)$ such that
$$\forall y\;\exists^{=1}z\;\forall x\;\...
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Looking for a source for Intended Interpretation
Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, ...
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Elementary proof of Chevalley's Theorem on constructible sets
I am looking for a proof of the easiest affine version of Chevalley's Theorem on constructible sets :
Theorem (Chevalley). The image of a constructible subset of $\mathbf C^n$ by a polynomial map $P:...
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Current Main Areas of Research in Model Theory [closed]
Could someone gives a general picture of the present state of Model Theory as a field? What are the current main areas and directions of research? What are some examples of the current experts and the ...
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Deciding isomorphism between graphs which interpret in the pure set
I am interested in the following decision problem:
Given descriptions of two graphs $G,H$ which interpret in the pure set $\mathbb N=(\{0,1,2,\ldots\},=)$, decide whether $G$ and $H$ are isomorphic....
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ω-categorical, ω-stable structure with trivial geometry not definable in the pure set
Briefly, my question is the following.
does every countable ω-categorical, ω-stable structure with
disintegrated strongly minimal sets interpret in the countable pure set?
By countable pure set I ...
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Examples of NIP fields of characteristic $p$
Definition. According to Shelah, a field $K$ does not have the independence property (i.e. is NIP) if for every first order formula $\varphi(x, \bar y)$ in the language of fields $(+,\times,0,1)$, the ...
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Reference request: Models of isomorphic languages result into isomorphic categories
This is basically a reference request by someone who has not been educated as a logician and would like to be rigorous about certain preliminary aspects of model theory.
Fix an uncountable universe $\...
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Hyperimaginaries and continuous logic
Classical (i.e., discrete) logic is well positioned to study imaginaries in part because the $T^{eq}$ construction allows us to treat imaginary sorts as we would treat any other sort. With ...
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When does an infinite model have a proper class-sized elementary extension?
Suppose that a set of sentences of a 1st order language has an infinite model $M$.
Under what conditions is there is a proper class-sized elementary extension of $M$?
How does the answer change if ...
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Riemann hypothesis in Zilber's field
Question. What is known about the situation (truth or falsity) of Riemann hypothesis in the Zilber's field?
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Constructive compactness for countable models?
The compactness theorem for countable (Tarski?) models is equivalent to the weak König's lemma by a result of H. Friedman and others as noted here, in the context of classical logic. The weak König's ...
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Compactness in Bishop's constructive mathematics
In Bishop's constructive mathematics, is there any literature on a possible version of the weak König's lemma, or of the compactness theorem for countable models? There is some related information ...
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Are there first order theories of interest to an algebraist or at least a model theorist of large cardinal consistency strength?
I am wondering if there are some first order theories of algebraic structures or structures of interest to model theorists of large cardinal consistency strength or at least unexpectedly high ...
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Intutionistic Robinson Arithmetic
By Friedman translation $HA$ and $PA$ prove the same $\Pi_2$ formulas.
Is it true for Intutionistic Robinson arithmetic(Robinson axioms with intutionistic logic) and classic Robinson arithmetic?
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What is the modern consensus on the difficulty of infinitesimals?
At a related thread at MSE an expert in reverse mathematics noted that "As the modern consensus is that only nonstandard models have infinitesimals, it will be quite challenging to give a concrete ...
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models of $I\exists^+_1$
$\phi$ is a $\exists^+_1$ formula iff it is in language of arithmetic and does not have $\forall$,$\neg$ and $\rightarrow$, therefore $I\exists^+_1$ is theory of $Q$+induction axioms for $\exists^+_1$ ...
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Do the analogies between metamathematics of set theory and arithmetic have some deeper meaning?
By "formal analogies" between the metamathematics of $\mathsf{ZFC}$/set theory and $\mathsf{PA}$(=Peano Arithmetic)/first order arithmetic, I mean facts such as the following:
We are considering a ...
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Variant of Conceptual Completeness
Let $\mathcal{C}$ and $\mathcal{D}$ be pretopoi, and let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a pretopos functor (that is, a functor which preserves finite coproducts, finite limits, and ...
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How to show certain theories are not existentially closed
To show that $ZFC$ is not existentially closed, we can use the following forcing argument: Let the ground model can model $V=L$ and the forcing extension model $2^{{\aleph}_{0}}=\aleph_{2}$. (Maybe ...