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 ...
mtg's user avatar
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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 ...
dorebell's user avatar
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3 votes
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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 ...
Dima Sustretov's user avatar
12 votes
3 answers
1k views

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 $\mathbb{P} \in M$, ...
dorebell's user avatar
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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?...
Joaquín Moraga's user avatar
5 votes
2 answers
469 views

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 ...
Alex Mennen's user avatar
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3 votes
1 answer
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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 ...
Noah Schweber's user avatar
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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 $...
Erfan Khaniki's user avatar
2 votes
1 answer
306 views

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 ...
Mikhail Katz's user avatar
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2 votes
1 answer
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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 $\Pi_1^0$ ...
Thomas Klimpel's user avatar
7 votes
1 answer
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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.) ...
Jonas Gomes's user avatar
2 votes
1 answer
183 views

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 ...
Alex Mennen's user avatar
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11 votes
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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 $\aleph_1$-...
Ioannis Souldatos's user avatar
8 votes
1 answer
332 views

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 $\...
namsap's user avatar
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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 $T$. ...
Noah Schweber's user avatar
4 votes
0 answers
174 views

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 ...
Kyle Gannon's user avatar
2 votes
2 answers
273 views

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 ...
Matt Brin's user avatar
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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 $(...
Alice's user avatar
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1 answer
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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 ...
urpzilmöräqÜ's user avatar
5 votes
1 answer
346 views

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 ...
urpzilmöräqÜ's user avatar
1 vote
1 answer
208 views

$\mathcal A\equiv\mathcal B\implies \mathcal A\cong\mathcal B$ 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\...
urpzilmöräqÜ's user avatar
3 votes
2 answers
214 views

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. ...
user42721's user avatar
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2 answers
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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, \...
Sam Roberts's user avatar
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3 votes
0 answers
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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 ...
TimZ's user avatar
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5 votes
2 answers
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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 ...
Alon Navon's user avatar
37 votes
3 answers
2k views

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 $...
Tim Campion's user avatar
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19 votes
0 answers
918 views

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 ...
Danielle Ulrich's user avatar
12 votes
0 answers
460 views

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 ...
喻 良's user avatar
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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 ...
6 votes
1 answer
631 views

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,...
Canonical User's user avatar
9 votes
1 answer
535 views

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\;\...
Domenico Zambella's user avatar
7 votes
6 answers
3k views

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, ...
Mikhail Katz's user avatar
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13 votes
1 answer
2k views

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:...
Drike's user avatar
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6 votes
0 answers
330 views

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 ...
Canonical User's user avatar
6 votes
1 answer
268 views

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....
Szymon Toruńczyk's user avatar
14 votes
2 answers
697 views

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 ...
Szymon Toruńczyk's user avatar
7 votes
2 answers
681 views

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 ...
Drike's user avatar
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1 vote
0 answers
194 views

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 $\...
Salvo Tringali's user avatar
7 votes
1 answer
568 views

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 ...
user avatar
5 votes
1 answer
343 views

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 ...
Haidar's user avatar
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10 votes
0 answers
507 views

Riemann hypothesis in Zilber's field

Question. What is known about the situation (truth or falsity) of Riemann hypothesis in the Zilber's field?
Mohammad Golshani's user avatar
5 votes
1 answer
406 views

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 ...
Mikhail Katz's user avatar
  • 15.1k
3 votes
1 answer
369 views

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 ...
Mikhail Katz's user avatar
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6 votes
1 answer
347 views

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 ...
Joseph Van Name's user avatar
13 votes
1 answer
661 views

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? ...
Erfan Khaniki's user avatar
3 votes
1 answer
1k views

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 ...
Mikhail Katz's user avatar
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1 vote
0 answers
87 views

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 $\exists^+_1$ ...
Erfan Khaniki's user avatar
42 votes
1 answer
2k views

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 ...
Gro-Tsen's user avatar
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23 votes
1 answer
2k views

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 ...
Jacob Lurie's user avatar
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2 votes
1 answer
162 views

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 ...
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