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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an algebraically closed field definable in a real closed field
Is it true that an algebraically closed field $k$ definable (in the model theory sense) in a real closed field $\mathcal R$ is the algebraic closure $\mathcal{R}^{alg}=\mathcal{R}(\sqrt{-1})$?
UPDATE:...
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Worst of both worlds?
It's well known that $\mathsf{AC}$ implies the existence of non-measurable sets. And it's also true that, if all sets are measurable, then $|\mathbb{R}/\mathbb{Q}| > |\mathbb{R}|$. But is there a ...
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Are these theories of real and complex number biinterpretable?
Let $T_R$ be the first-order theory of real closed fields. This is precisely the theory over the language $\{0,1,+,\times\}$ such that the theorems are the formulas that hold in $\Bbb R$. It can be ...
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What is the lowest complexity definition of $\mathbb{Z}$ in an infinite extension of $\mathbb{Q}$
In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park proved a result which implies that $\mathbb{Z}$ is $\exists\forall$-...
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Does Keisler-Shelah isomorphism theorem hold for infinitary logics?
In model theory, the Keisler-Shelah isomorphism theorem asserts that two models of a theory are elementary equivalent if and only if they have isomorphic ultrapowers. On the other hand, assuming that $...
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If two structures are elementarily equivalent, is there a zigzag of elementary embeddings between them?
Fix a first-order signature $\Sigma$. There is an equivalence relation $\sim$ on the class $\Sigma\mathrm{-Str}$ of all $\Sigma$-structures given by $M \sim N$ iff $M$ and $N$ are elementarily ...
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models of PA which are isomorphic but not elementarily equivalent?
On page 164 of his book Models of Peano Arithmetic, Kaye states Friedman's Theorem:
Let $M{\vDash}PA$ be nonstandard and countable, let $a\in M$ and let $n\in {\mathbb N}$. Then there is a proper ...
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Automorphisms of projective spaces, and the Axiom of Choice
It is known that upon not accepting the Axiom of Choice (AC), there exist models of ZF in which there are projective spaces (over a division ring) with a trivial automorphism group. (This is a truly ...
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How big is the least non-$\Sigma^1_1$-pointwise-definable ordinal?
There's a large countable ordinal which has cropped up (as a lower bound!) in a computable structure theory problem I'm playing with. At present I don't really understand how big it is, and I'm ...
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Concept of bedrock and mantle in the multiverse view in the philosophy of mathematics
To be clear, I am not a mathematics educated student and I can not follow the details of the technicality of the forcing extension, but I feel that I have a good understanding of the big picture (of ...
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Is there a stable structure on $[0,1]$ that approximates every continuous function?
The $n$-dimensional form of the Weierstrass approximation theorem is the statement that polynomial functions are dense under the $\ell_\infty$-norm in the space of continuous functions on $[0,1]^n$ ...
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Models of arithmetic in a signature with exponentiation but not addition and multiplication
Let $\mathcal{L}_{\mathrm{exp}}$ be the language with signature $(0, ^\prime, <, \mathrm{exp})$ (with $0$ interpreted as zero, $^\prime$ as successor, and $\mathrm{exp}(x)$ as $2^x$) and let $\...
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Elementary extensions of direct product
I apologize if this question is elementary: Let $A$, $B$, and $A^{\prime}$ be groups such that $A^{\prime}$ is an elementary extension of $A$. Is it true that $A^{\prime}\times B$ is an elementary ...
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Are the definable hyper-reals, using quantifiers only over the standard reals and natural numbers, the same as the algebraic numbers?
This question arose today at Yevgeny Gordon's talk, "Will nonstandard analysis be
the analysis of the future?" at the CUNY Logic
Workshop. Here is my way of asking it.
Consider the ordered real field ...
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Non-normal numbers definable without parameters in the langauge of differential rings with composition
Background: It is currently unknown whether $e$ is normal. A natural way to approach this question is to find a class to which $e$ belongs, and prove all members of that class are normal. For example, ...
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Expressive power of FO with $\mu$
Let us consider the first-order logic extended with the least fixed point operator (FO+LFP). That is, together with the usual first-order formulas, we also have formulas of the form:
$$\mu X[\...
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Applications of "model-theoretic" forcing
The notion of forcing was invented by Paul Cohen, who used
it to prove the independence of the Continuum Hypothesis. He constructed a model of set theory in which the CH fails, thus showing that CH ...
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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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Self-defining structures
The relations $R$ in abstract graphs (with genuinely propertyless vertices) cannot be defined because there is nothing the relations can base on: they have to be presupposed.
But consider derived ...
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Does "agreement on cardinalities" imply second-order elementary substructurehood?
Say that a logic $\mathcal{L}$ satisfies the weak test property iff for all $\mathfrak{A}\subseteq\mathfrak{B}$ we have $(1)\implies(2)$ below:
For each $\mathcal{L}$-formula $\varphi$ with ...
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The lattice of analogues of Robinson's $Q$
This question was asked and bountied at MSE without response.
Call a sentence $\varphi$ in the language of arithmetic $Q$-like iff $\mathbb{N}\models\varphi$ and $\{\varphi\}$ is essentially ...
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Iterating definability
An odd -- probably basic -- question about model theory:
For $\mathcal{M}$ a structure in a (first-order) signature $\Sigma$, let $\mathcal{M}'$ be the structure in signature $\Sigma\sqcup\lbrace U\...
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Is $Ded(\kappa)<Ded(\kappa)^\omega$ consistent?
Hello,
I want to ask if anyone can tells us what is known (consistently) about $Ded(\kappa)$, $\kappa$ an infinite cardinal.
Definition If there is a dense linear order w/o endpoints of size $\...
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Consistent hierarchy of axiomatic systems
First of all, I am not an expert in model theory. I just want to get my personal view on the foundations of mathematics straight.
I just learned in Sergey Melikhov's answer to another question ...
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Paris-Harrington principles parametrized by functions $f:\mathbb N \to \mathbb N$
Recall that the Paris-Harrington Principle, $\mathsf{PH}$, is the statement that for each $e, r, k < \omega$ there is an $N < \omega$ so that given any coloring $c:[N]^e \to r$ there is an $H \...
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Order types of models of theories of ordinals
For $C$ a set of ordinals, let $\mathcal{L}(C)$ be the language with identity, a relation symbol for less than, function symbols for successor, addition, multiplication, and exponentiation, and a ...
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Does non-stablity imply that there is a difference between non-forking and coheir extension
Fix some theory $T$.
Let $p$ be a type over some Model M and let $q$ be some global extension of $p$.
Note:
The number of global coheirs of $p$ is bounded by the number of ultrafilters on $M$.
Also ...
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Where's the notion of interpretation (model) originally introduced?
Where's the notion of interpretation (model) originally introduced?
I find it used in Skolem's paper "Logico-combinatorial investigations in the satisfiability or provability of mathematical ...
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Is there a largest o-minimal structure all of whose definable functions are analytic?
In the paper "Quasianalytic Denjoy-Carleman classes and o-minimality" by J.-P. Rolin, P. Speissegger and A. J. Wilkie, it is proven that there is no largest o-minimal structure on the real ...
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Modal logic of "mostly-satisfiability"
For $n\in\omega+1$ let $\mathsf{ZFC}_n$ be $\mathsf{ZC}$ + $\{\Sigma_k$-$\mathsf{Rep}: k<n\}$. Let $\widehat{\mathsf{ZFC}}$ be the strongest consistent theory $\mathsf{ZFC}_n$ (so if $\mathsf{ZFC}$ ...
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On the strength of higher-logic analogues of $\mathsf{ZFC}$ + Montague's Reflection Principle
Throughout, I work in $\mathsf{MK}$ in order to be able to conveniently quantify over logics; if one prefers, we can restrict attention to (say) $\Sigma_{17}$-definable logics and work in $\mathsf{ZFC}...
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Is there an abstract logic that defines the mantle?
It is a known result by Scott and Myhill that the second-order version of $L$ yields $\mathrm{HOD}$.
Recently, Kennedy, Magidor, and Väänänen (Inner models from extended logics: Part I and II) ...
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Where do nonstandard elliptic curve angles come from?
This is a question which has bounced around my head over the past few years. At the same time, I am answering Riemann hypothesis for zeta function of algebraic curves over fields of infinite ...
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Intuition behind Boolean-valued models of set theory
$\DeclareMathOperator\Card{Card}$The book Forcing Eine Einführung in die Mathematik der Unabhängigkeitsbeweise by Hoffmann provides an intuition behind boolean valued models of set theory which I will ...
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"Local" compactness properties beyond $\mathcal{L}_{\omega_1,\omega}$?
Below, all languages are finite for simplicity.
This question is about generalizations of Barwise compactness for logics more complicated than $\mathcal{L}_{\omega_1,\omega}$: properties of the form "...
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How strong is exponentiation with only open induction? (Or: "how low can we go?")
Do the strongest theories currently known to be unconstrained by Tennenbaum's theorem ($IOpen$ and some modest extensions) remain so when augmented with a definition of exponentiation and axiom $\...
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Is the hypotenuse operation associative in every Tarski plane?
By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\...
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First order formula describing connected components
I ask this question here after no answer came up in the original MathSE question.
Let $\mathcal{L}$ be the language $\{+,-,\cdot,0,1,P\}$ where $P$ is some $n$-ary relation symbol. Is there a formula $...
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Does determinacy imply unravellability for the Borel sets (over a weak base theory)?
As far as I know, the only way we currently know how to prove Borel determinacy in $\mathsf{ZFC}$ is to go through unravelability (a rather technical property whose definition can be found in Martin's ...
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Topological Vaught's conjecture for special theories
As is know, Vaught's conjecture is a special case of topological Vaught's conjecture.
On the other hand, the Vaught's conjecture is true for the following theories:
1- $\omega$-stable theories (...
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Monadic second-order theories of the reals
I’m looking for a survey of monadic second-order theories of the reals.
I’m starting from a 1985 survey by Gurevich which says (p 505) that true arithmetic can be reduced to “the monadic theory of ...
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Is there any theorem achieving Conway's "Mathematician's Liberation Movement"
John Conway on in the appendix to part zero of ONAG describes a "Mathematician's Liberation Movement". The goal would be to give mathematicians the freedom to create mathematical theories with the ...
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On undefinability of well-orderings in $L_{\omega_1,\omega}$
It is a well-known theorem that well-orderings can not be characterized in $L_{\omega_1,\omega}$.
In particular, if $\psi$ is an $L_{\omega_1,\omega}$-sentence in a vocabulary $\tau$ that contains a ...
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Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$?
Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$?
In End extensions of models of linearly bounded arithmetic paper the author said this problem is open. I want ...
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Kripke models of $HA$
Let $K$ be a kripke model and $k$ be one of its node, then $\mathcal{M}_k$ is classical structure of $k$.
What is the strongest theory of arithmetic like $T$ such that for
every kripke model $K\...
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What classes of complex manifolds are known to be definable in an o-minimal expansion of the real field?
It is a widely known (perhaps slightly folkloric) fact that compact complex manifolds, understood as first-order structures with a predicate for each analytic subset, are definable in an expansion of ...
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Can the isomorphism relation for countable models become harder when adding finitely many constants?
I am particularly interesting in the case where $T$ is o-minimal, but I would be interested in any theory $T$ (or even an $L_{\omega_1,\omega}$-sentence) which has this property.
Context: view the ...
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Why is definable compact equivalent to bounded and closed for sets with o-minimal structures?
We have $M$ an o-minimal structure. $X \in M^n$ with the induced topology.
I'm reading an article which shows that $X \in M^n$ is definable compact is
equivalent to $X$ being bounded and closed.
...
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Model ${\sf ZF}$ that "spreads" members of ${\cal P}(X)$
Is there a model of ${\sf ZF}$ such that there is an infinite set $X$ and a injective map $f:{\cal P}(X)\to {\cal P}(X)$ so that for $a\neq b \in {\cal P}(X)$ we have $|f(a)\cap f(b)| \leq 1$?
Note. ...
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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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