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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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A new generalization of the dimension?

During my research, I came a cross on these notions : Definition 1: A structure $S$, is a pair $(X, T)$ with $X$ a set and $T$ a set of subsets of $X$, stable by arbitrary intersections, with $X \...
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Differences between logic with and without equality

By logic without equality I mean those kind of logics where equality is treated as a binary relation satisfying some axioms, as opposed to a logics where equality is a logical symbol satisfying some ...
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The surreal numbers under a change of universe

Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\...
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A peculiarity of Henkin's 1950 proof of completeness for higher order logic

My question concerns Henkin's original (1950) completeness proof https://projecteuclid.org/euclid.jsl/1183730860 for classical higher order logic and type theory relative to so-called general models. ...
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Is there a pair of non-isomorphic structures each of which is isomorphic to an ultrapower of the other?

Does there exist a pair of non-isomorphic structures $\mathfrak{A}$ and $\mathfrak{B}$ as well as sets $I$ and $J$ and ultrafilters $\mathcal{U}$ on $I$ and $\mathcal{F}$ on $J$ such that $\mathfrak{A}...
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O-minimality and forcing

It is well-known that the structure $(\mathbb{R}, +, \cdot, <, 0, 1)$ is an o-minimal structure and hence the set of integers $\mathbb{Z}$ is not definable in it. In an ongoing project with Will ...
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Is there an expansion of $(\mathbb{N},+,<)$ by a pairing function that is still NIP?

I was told once that there is a theory consisting of just a pairing function that is stable, although I cannot find a reference for it. This motivated my question, which is essentially the title, ...
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Definable functions in $o$-minimal structures

Consider the field of real numbers $(\mathbb R,+,\cdot)$. An expansion of $(\mathbb R,+,\cdot)$ is a tuple $(\mathbb R,+,\cdot,S)$, where $S$ is a collection of subsets of $\mathbb R^n$ for $n \in \...
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If any satisfiable $\mathcal{L}_{κ,κ}(Q_{=κ})$-theory remains satisfiable when replacing $Q_{=κ}$ with $Q_{=μ}$, is $κ$ huge?

Recently, I have asked a model-theoretic question concerning a weakening of different forms of compactness. I now present another model-theoretic question as a weakening of hugeness. If any ...
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Functoriality of indiscernible sequences

Let $T$ be a first order theory of, say, some type of combinatorial geometries which contain indiscernible sequences of points. Let $(\Gamma,\mathcal{O})$ be a model of $T$, where $\Gamma$ is the ...
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Self-additive posets

We say that a partially ordered set $(P,<)$ is self-additive if the two natural embeddings of $P$ in $P\oplus P$ (the linear sum of $P$ and itself) are elementary. We have the following. ...
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A weakening of cardinal compactness - is it equivalent?

I was messing around with the intuition behind the size of weakly compact cardinals (in their usual characterization). I found an interesting, seemingly weaker LCA which still implies weak ...
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Looking for help in defining a new epistemic logic

I'm looking for some guidance in defining a new epistemic, temporal logic. I am looking to extend a logic called Sequential Epistemic Logic (SPAL): https://pdfs.semanticscholar.org/dae6/...
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Proof of the Specker-Blatter theorem

The theorem in the title states the following: If $\mathcal{C}$ is a class of structures definable in monadic second order logic with unary and binary relation symbols only, then the function $f_\...
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Does this consequence of measurability in terms of games of length $\omega+1$ imply measurability?

For any two structures $\mathcal{M}$ and $\mathcal{N}$ in the same first-order language $\mathcal{L}$ and any ordinal $\theta$, let $G_\theta(\mathcal{M},\mathcal{N})$ be the two-player game of ...
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1answer
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Partition theorems for located words

In this paper Bergelson, Blass, and Hindman prove the following Theorem 1.2 Let $W(\Sigma; v)$ be colored with finitely may colors and let $\bar s$ be an infinite sequence from $W(\Sigma; v)$. ...
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Is any real closed extension of $\mathbb R$ characterized up to isomorphism by its ladder?

Let $R$ be a real closed field. Recall that the ladder of $R$ is the divisible, ordered abelian group obtained by quotienting $R$ by a certain equivalence relation. Note that $R$ has trivial ladder ...
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theories where angles exist without a metric

The underlying basic question, which I'm sure I'm not the first to ask, is what are the possible exotic/nonintuitive models of Euclid's axioms/postulates, outside the one where "lines" are interpreted ...
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2answers
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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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684 views

Set theory bootstrapping

Let $\mathcal{L}$ be the first order language of ZFC set theory, and let $\mathcal{L}_{\infty,\infty}$ be the usual infinitary extension of the language allowing arbitrary long disjunctions/...
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Model theory of Banach algebras

Let us consider the (metric) theory of Banach algebras. I have a sentence encoding the (possible) openness of multiplication in a given Banach algebra: $$(\forall x) (\forall y) (\forall \varepsilon &...
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Importance of Simple Theories

In model theory, a complete first-order theory $T$ is said to be simple if each type does not fork over some subset of its domain of size at most $|T|$. Question (1). What is the significance of ...
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Generic two-cardinal behavior of first-order sentences

This is a hopefully improved version of a question I asked before and then deleted because it was based on some fundamentally incorrect assumptions. Some first-order theories are able to control the ...
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1answer
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For which theories does ZFC without global choice prove the existence of a proper class monster model?

Proper class sized monster models are typically formulated in a class theory like $NBG$ and they can reasonably be formalized in $ZFC$ with some kind of global choice, but for some theories you don't ...
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What is the interest of Grothendieck ring in Model theory

I've hear of the notion of Grothendieck ring in model theory. Could someone tell me what are their interest and what are the relevant questions about them ? I'm sorry for this quite vague question.
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Forall exists formula

This is a reference request. Let $A$ be a formula in the language of rings which is of the form $\forall_{x_1}\dots\forall_{x_n}\exists_{y_1}\dots\exists_{y_m} F$, where $F$ is quantifier-free. I once ...
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Elementary equivalence of monoidal categories =?

Recall that, in model theory, two models $M_1$ and $M_2$ of the same signature are elementary equivalent if $ M_1 \models \phi \Leftrightarrow M_2 \models \phi $ for every first order formula $\phi$ ...
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Can the number of countable models of a complete first-order theory decrease after adding constants?

If $T$ is a countable complete first-order theory with infinite models, the number of countable models it has, $I(T,\omega)$, must be an element of $N=\{1,3,4,5,6,7,\dots,\omega,\omega_1,2^\omega\}$ (...
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How to extract an indiscernible sequence from a set in a stable theory?

I was reading Chernikov's notes about stable theories, and he mentions the following fact: If $T$ is stable and $A$ is some set of parameters large enough then there is some indiscernible sequence $I ...
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How to construct “inaccessible hypernatural”?

Consider that, take a sufficient large natural number $a_1$, then take a natural number $a_2$ sufficient large to $a_1$, then take $a_3$,... Now we have a function $n \mapsto a_n$ which grows very ...
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Can an uncountable model of Peano Arithmetic be recursive?

Can an uncountable model of Peano Arithmetic be recursive? What does it mean for an uncountable model to be recursive? Well, we represent the elements of the model using real numbers instead of ...
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Stability and complete types (in Model Theory)

I read the following statement in these slides of Saharon Shelah: "$K$ is stable iff for every $M \in K$ there are only "few" complete types over $M$." About the notation: here $K$ consists of all ...
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Which large cardinals have a Matryoshka characterization?

What on Earth do Russian Matryoshka dolls have in common with large cardinal axioms?! Well, the answer lies in Jónsson algebras! Here is how: As illustrated in the pictures, a Matryoshka set is a ...
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Are all generalized Scott sets realized as generalized standard systems?

Below, I've focused on PA when lots of other theories would do. If replacing PA with a different theory leads to a more answerable question, feel free to do so. The standard system of a nonstandard ...
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1answer
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Is there a Baldwin-Lachlan style characterization of countable unidimensional theories?

It's well known that a countable theory is uncountably categorical if and only if it is $\omega$-stable and has no Vaughtian pairs. One of the old definitions of unidimensional theory (all $\omega_1$-...
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On models of $Th_{\Pi_2}(PA)$

Let $M$ be a nonstandard model of $PA$. Q1. Is there any way to get a submodel $N\subset M$ such that $N\models Th_{\Pi_2}(PA)$, but $N\not\models PA$? Q2. Especially, what combinatorial ...
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How are incompleteness and independence proofs related?

(1) Some typical inclompleteness proofs use a kind of fixed point argument - for certain $\Phi$ you find a $\varphi$ with $\Phi(\mathrm{code}(\varphi))\leftrightarrow\varphi$. (2) Some independence ...
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Self homomorphisms of hyperreals fixing the reals

What do we know about the circumstances (whether having to do with the axioms of set theory or the model itself) under which a field $F$ of hyperreals (=ultrapower of $\mathbb R$ with respect to a non-...
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$\omega_1$-like elementary chains from long countable elementary chains

For some countable first-order theory $T$, if we have a linearly ordered set $I$ and an elementary chain $\{\mathfrak{A}_i\}_{i\in I}$ we can form a structure $\mathfrak{A}_{I}^\star = (\bigcup_{i\in ...
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What conditions make this embedding become algebraically closed?

This question is just out of curiosity. Let $\mathcal{L}$ be the first-order language of theory of rings. Let K and F are two fields such that $K \subseteq F$. If K is existentially closed in F, ...
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What “metatheory” did early set theory/logic researchers use to prove semantic results?

Things like the first-order completeness theorem and the Löwenheim-Skolem theorem are considered foundational in mathematical logic. The modern approach seems to be, usually, to interpret a "model" ...
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Countable elementary sub-structure of the automorphism group of the binary rooted tree

Let $G$ be the automorphism group of the binary rooted tree. The downward Löwenheim-Skolem theorem states that G has a countable elementary sub-structure. My question is whether such sub-structure ...
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1answer
251 views

Definability in the field of reals with a predicate for some powers of two

In "The field of reals with a predicate for the powers of two", Van den Dries has proved that the set of integers is not definable in $(\mathbb{R}, +,\cdot, \leq, 0, 1, 2^{\mathbb{Z}})$, where $2^{\...
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1answer
323 views

Axiomatizable $\exists \forall$ theory

I have been thinking the following problem proposed by my friends for a long time. Let $\mathcal{L}$ be the first-order language of theory of rings and let $K$ be the class of algebraic number ...
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CH and automorphisms of ultrapowers of $\mathbb{Z}$ and $\mathbb{R}$

Notation and motivation. Given an algebraic structure $\mathbb{M}$ of cardinality at most the continuum and with countably many operations, and a nonprincipal ultrafilter $\cal{U}$ on a countably ...
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Are the rationals definable in any number field?

Let $K$ be a number field. Is it necessarily true that $\mathbb{Q}$ is a first-order definable subset of $K$? Equivalently (since in any number field, its ring of integers is a definable subset), is $\...
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1answer
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When are pseudofinite theories almost sure theories?

Let $T$ be a complete theory (say in a countable language, though this may not be required). Assume that $T$ is complete with infinite models. Question 1) When is the theory of $T$ an almost sure ...
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Analog of Tennenbaum's theorem for EFA

EFA can prove the exponential function to be total, but it cannot prove the superexponential function to be total. Is there an analog of Tennenbaum's theorem (which states the PA has no recursive non-...
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What are the requirements of a foundational theory?

There are multiple languages to describe all of mathematics, and there are some equivalences between them, some more successful then others. My question is can we describe some requirements (in some ...
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Is there a restriction of Linear Temporal Logic that has a “Markov” property?

I have a problem, that I can formulate as model-finding in Linear Temporal Logic (via Büchi automata). I also have the additional knowledge, that there is always satisfied a Markov-like property, ...