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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Stabilization of recursive approximation in $PA^-+I\Sigma_1^0$

Over any model M of $PA^-+I\Sigma_1^0$. Suppose $A\in [T]$ where $T$ is a $\Delta_2^0$-tree and $A$ is one isolated path. Further, $A$ is regular, i.e. $\forall n A\upharpoonright n$ has a code in $M$....
Jing Zhang's user avatar
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Algebras admitting quantifier elimination

I apologize if this question is meaningless or trivial: What are examples of Algebras admitting quantifier elimination? Especially are there Groups admitting quantifier elimination? I need to say ...
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A preprint of Sela concerning the work of Kharlampovich-Miyasnikov

Yesterday, Z. Sela published a preprint in arXiv which claims that the solution of Olga Kharlampovich and Alexi Miyasnikov for the Tarski problem on decidablity of the first order theories of free ...
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Negated varieties and their relatively free algebras

During the past days, I asked some questions in order to gain a clear understanding of the notion of "free algebras". I suppose that the question below is the most clear image of the concept I have ...
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The existence of an algebra whose set of identities and first order theory are equivalent

Is there an algebra $A$ (for example a group) such that $Th(A)$ is logically equivalent to $id(A)$? In other words, is there an algebra $A$ such that $$ Mod(Th(A))=Var(A)? $$ Clearly finite algebras ...
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relatively free groups in $Var(S_3)$

Suppose $S_3$ is the symmetric group of order 6. Which elements of the variety $Var(S_3)$ are relatively free? This question is related to my previous question Relatively free algebras in a variety ...
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Universal graphs on higher cardinals

The Rado graph contains every finite graph as induced subgraph, and its also holds for countable graphs. So it is an universal graph of size $\aleph_0$, which contains all graphs of size $\aleph_0$ as ...
István Kovács's user avatar
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Stability of analytic Zariski structures

Noetherian Zariski structures are introduced by Hrushovski and Zilber(1996). An analytic Zariski structure is a generalization of Noetherian Zariski structures, introduced by Zilber and Peatfield. ...
Mostafa Mirabi's user avatar
11 votes
1 answer
346 views

Is "approximate categoricity" absolute?

Let $T$ be a countable first-order theory, and assume that $T$ has exactly one atomic model up to isomorphism in every uncountable cardinality. (By "atomic" I mean a model which omits the non-...
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Peano (Dedekind) categoricity

What is the smallest fragment of second order logic such that $Th(\mathbb{N})$ in that logic is categorical (only one model, namely natural numbers, up to isomorphism). For example, can we do this in ...
Thinniyam Srinivasan Ramanatha's user avatar
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Recursive ordinals and the minimal standard model of ZF

Does the minimal standard model of ZF contain all recursive ordinals or is it limited (probably by the proof theoretic ordinal of ZF as I suspect but cannot prove)? Paul J. Cohen's definition of the ...
Paul Budnik's user avatar
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Recursive Non-standard Models of Modular Arithmetic? [closed]

Any algebraically closed field (ACF) is a model of Modular arithmetic (MA). (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)...
Russell Easterly's user avatar
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1 answer
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Outer Definability of a Class

Definition: Let $C$ be a class of sets and $\mathcal{L}$ a first order relational language. We say $C$ is "outer definable" by $\mathcal{L}$ if there is a first order theory $T$ and for each $n_{R}$ - ...
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Is there a 0-1 law for the theory of groups?

Several months ago, Dominik asked the question Is there a 0-1 law for the theory of groups? on mathstackexchange, but although his question received attention there is still no answer. By asking the ...
Seirios's user avatar
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Generalizations of Birkhoff's HSP Theorem

Let $\mathbf{C}$ be the class of algebraic structures of some fixed type satisfying some sentence $\phi$. Birkhoff's HSP theorem says that $\mathbf{C}$ is closed under homomorphisms, subalgebras and ...
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$(\kappa,\lambda)$ - Minimal Models & Stronger Version of Rowbottom's Theorem

Definition 1: Let $M$ be a $\mathcal{L}$ - structure and $A\subseteq Dom(M)$. Define: $Def_{A}(M):=\{X\subseteq Dom(M)~|~\exists n\in \omega~~\exists \varphi (x,y_1,...,y_n)\in \mathcal{L}-Form~~\...
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3 votes
1 answer
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$(\kappa , \lambda)$ - Minimal Models of $\text{ZF}$

The notion of minimality in model theory is related to the existence of a gap in the size of definable subsets of a model. Now consider the following generalization: Definition 1: Let $M$ be a $\...
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5 votes
1 answer
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From elementary equivalence to isomorphism

A few years ago, when I took the basic course in Logic, I was very surprised to discover that given a signature $\sigma$ and two structures $M$ and $N$ of $\sigma$ which are elementarily equivalent (...
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Joint Forcing Extension Property

Definition 1: A class $\mathcal{K}$ of countable transitive models of $\text{ZF}$ has strong "joint forcing extension property" (JFEP) iff for all $M,N\in \mathcal{K}$ there are forcing notions $\...
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2 votes
1 answer
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Expressing "There are more Fs then Gs" in first-order logic

Is there a sentence of first-order logic that's true in all finite interpretations in which there are more Fs than Gs, and false in all finite interpretations in which there are not more Fs than Gs? ...
GeorgeAD's user avatar
8 votes
0 answers
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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 ...
Dima Sustretov's user avatar
3 votes
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162 views

Cardinality of Grothendieck ring in model theory

Good evening, In model theory there is a notion of Grothendieck ring defined here http://math.berkeley.edu/~scanlon/papers/greu12jun00.pdf. Do we know anything about the cardinality of these rings ? ...
guest's user avatar
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1 answer
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Shelah's book on "Classification Theory"

As we know one of the most important and fundamental books in stability, simplicity, forking and ... classification theory, is Shelah's "Classification Theory" where lots of original ideas of the ...
Ali V.'s user avatar
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3 answers
647 views

Turing Functional and $\Sigma_1^0$-formulas in models of fragments of PA

In models of PA with restricted induction power (for example, only $I\Sigma_n$ is present), the failure of higher induction scheme is characterised by the existence of definable cuts (like $\Sigma_2$ ...
Jing Zhang's user avatar
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15 votes
3 answers
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The interplay between certain aspects of interpretability, model theory and category theory

I have some questions about the interplay of interpretability, model theory and category theory. Since I had difficulties in finding literature or other helpful information about this topic, it would ...
user avatar
7 votes
1 answer
460 views

Vaught conjecture for uncountable languages

Recall Vaught conjecture: the number of countable models of a first-order complete theory in a countable language is finite or $\aleph_0$ or $2^{\aleph_0}.$ Now let $\lambda$ be an uncountable ...
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7 votes
1 answer
349 views

Is the consistency of $\mathcal{L}_{\infty\omega}$-sentences absolute?

The question is exactly that of the title. Suppose $\varphi\in V$ is an $\mathcal{L}_{\infty\omega}$-sentence, and $W$ is an inner model of $V$ such that $\varphi\in W$. Is the statement $\varphi$ ...
Noah Schweber's user avatar
7 votes
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552 views

Recent application of model theory in set theory by Shelah-Malliaris

Recently one of the oldest open problems in set theory about the cardinal invariants of the continuum (i.e the question of whether $\mathfrak{p}=\mathfrak{t}$) was solved by Shelah and Malliaris (see "...
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2 votes
1 answer
317 views

Is elementary equivalence absolute?

Assume we have two objects $M_1$ and $M_2$ models of respective $L_{\omega_1,\omega}$-sentences $\Sigma_1$ and $\Sigma_2$. Assume $M_1$ and $M_2$ are elementarily equivalent in some model of set ...
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12 votes
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Is there a Rado category?

The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between finite/...
Qfwfq's user avatar
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4 votes
1 answer
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Sketches for categories of models of complete theories

In Accessible categories : the foundations of categorical model theory, chapter 3 p.58, Makkai and Paré claim that there is "an (obvious) identification of a class of sketches so that the categories ...
Sylvain's user avatar
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1 answer
310 views

Completeness theorem via syntactic categories

The nLab says in its internal logic article that the Completeness Theorem can be proven via a ``generic model'' of the theory. The model is generic in the sense that the only things true of it are ...
Luke's user avatar
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9 votes
1 answer
650 views

Does existence of a proper class model imply the consistency?

The fundamental theorem of model theory says that: Theorem: A first order theory is consistent if and only if it has a model. In the above theorem we assume that the domain of any model is a non-...
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3 votes
2 answers
537 views

Can we force with Fraisse filters to solve Vaught's conjecture?

Around the classic Fraisse amalgamation theorem in model theory we have the following notions: Definition (1): If $M$ be an $\mathcal{L}$-structure then define: $age(M):=\lbrace N~|~N~\text{is ...
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1 vote
0 answers
254 views

Is there a non-trivial consistency preserving transformation?

In ‎set ‎theory ‎"equiconsistency" (and not "consistency") ‎of ‎the ‎theories ‎is the‎ ‎main ‎part ‎of ‎researches. ‎So ‎we ‎usually ‎try ‎to ‎construct a‎ ‎new model ‎using a‎ ‎given ‎one. ‎In ‎the ‎...
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5 votes
1 answer
299 views

What are the Possible Large Cardinals of $L[X]$?

I've been doing some basic reading in inner model theory, and I'm at the point where I've seen the definition of things like Martin-Steel and Mitchell-Steel inner models. I am wondering about the ...
user40919's user avatar
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7 votes
1 answer
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Shelah's categoricity conjecture

Does Shelah's categoricity conjecture for abstract elementary classes have applications in other branches of mathematics?
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14 votes
4 answers
2k views

Where is the end of universe?

In some sense the empty set ($\emptyset$) and the global set of all sets ($G$) are the ends of the universe of mathematical objects. The world which $ZFC$ describes has an end from the bottom and is ...
user avatar
6 votes
1 answer
214 views

Absoluteness of completeness

Suppose $V_0, V_1$ are (not necessarily well-founded) models of ZFC and suppose $\varphi$ is a first order sentence in a finite language $L$ (in our background model of set theory). Because every true ...
Nate Ackerman's user avatar
7 votes
1 answer
282 views

Is there a forcing closure?

The main theorem of forcing says that for any c.t.m of $ZFC$ like $M$ and for all partial order $\mathbb{P}$ and $\mathbb{P}$-generic $G$ over $M$, there is a c.t.m of $ZFC$, like $N$ such that $N$ is ...
user avatar
10 votes
1 answer
1k views

Are all complete finitely axiomatizable first order theories $\aleph_0$-categorical?

Suppose $T$ is a complete first order theory with a finite axiomatization. Must $T$ be $\aleph_0$-categorical? If not are there any simple examples of finitely axiomatized complete first order ...
Nate Ackerman's user avatar
1 vote
0 answers
863 views

Relativization of Formulas and Models [closed]

I want to show that the definition of satisfiability is consistent with the definition given by relativization, i.e. Let $L=\{\in\}$. Let $M$ be a definable set and let $E\subset{M\times{M}}$. Let $\...
UserB1234's user avatar
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3 votes
1 answer
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Do inner models of unique measurable cardinals have a regular behavior? (Edited and Revised Version)

We know that if ‎$‎‎\kappa‎‎$ is a‎ ‎measurable ‎cardinal ‎and ‎‎$‎‎\mu$ be a‎ ‎two-valued ‎non-trivial‎‎$‎‎‎\kappa‎$-additive ‎measure ‎on ‎it ‎the‎n the corresponding inner model produced by ...
user avatar
7 votes
2 answers
356 views

Is there an inner model between two distinct inner models of ZFC?

Definition (1): A‎n‎‎ ‎inner ‎model ‎of ‎‎$‎ZFC‎$ ‎is a‎ ‎tarnsitive proper class ‎model ‎‎of $‎‎ZFC$ ‎which ‎contains ‎all ‎ordinal numbers. ‎Informally ‎we ‎denote ‎the ‎collection ‎of ‎all ‎inner ‎...
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39 votes
4 answers
4k views

Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic

Mathematician Edward Nelson is known for his extreme views on the foundations of mathematics, variously described as "ultrafintism" or "strict finitism" (Nelson's preferred term), which came into the ...
Keshav Srinivasan's user avatar
7 votes
1 answer
714 views

What are the essential properties of algebraic closure on an arbitrary structure?

Define the "model theoretic" notion of a closure function as follows: Definition (1): Let $D$ be a non-empty set. A function $cl:P(D)\longrightarrow P(D)$ called a closure function iff it has the ...
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32 votes
3 answers
2k views

Is the field of constructible numbers known to be decidable?

By the field of constructible numbers I mean the union of all finite towers of real quadratic extensions beginning with $\mathbb{Q}$. By decidable I mean the set of first order truths in this field, ...
Colin McLarty's user avatar
2 votes
1 answer
276 views

Can we flex the rigid models by enough power?

Definition (1): ‎An ‎‎$‎‎‎\mathcal{L}‎$ -‎ ‎structure ‎‎$‎‎‎\mathcal{M}‎$ ‎called ‎"‎‎rigid" ‎iff ‎‎there ‎is ‎no ‎non-trivial automorphism on ‎$‎\mathcal{M}‎$.‎ ‎‎ Definition (2): ‎An ‎‎$‎‎‎\mathcal{...
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6 votes
1 answer
626 views

Omitting types and Baire category

What is the relation between omitting types theorems in model theory and the baire category theorem?
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11 votes
2 answers
563 views

Extending a partial order while preserving an automorphism

It is well known that if $(P, \leq)$ is a partial order then $\leq$ can always be extended to a linear order. This is sometimes called Szpilrajn´s theorem although it had been previously proved by ...
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