Model theory is the study of (classes of) mathematical structures (e.g. groups, fields, graphs, universes of set theory) using tools from mathematical logic. Objects of study in model theory are models for formal languages which are structures that give meaning to the sentences of these formal ...

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Uncountable models for integers

Part of Asaf Karagila's brilliant answer to one of my other questions puzzles me a lot. Namely, I find it hard to understand how there can be a model for ZFC with uncountably many integers. My ...
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1answer
109 views

Maximal model for $\Bbb R$?

I have not dealt professionally with set theory, so excuse me if my way of formulating this question does not completely follow standard terminology. Actually, my question is about whether or not the ...
4
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103 views

Relations between Theories and Categories

I'm just toying around with some thoughts, trying to grock some concepts: It seems that every formal theory induces a locally small category via interpretations: its objects are structures that ...
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43 views

When a subgroup of automorphism group of a structure is in the form of automorphism group of a substructure?

Question 1: Is the following statement true? ($*$) Let $\mathcal{L}$ be a first order language and $\mathcal{M}$ a $\mathcal{L}$-structure and $H\leq Aut(\mathcal{M})$ then there exists a ...
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194 views

Is “PA has no non-standard models” consistent with ZF?

I have seen several proofs that there exist nonstandard models of arithmetic, but they all seem to rely on the compactness theorem, which is not implied by ZF. So are there any proofs in ZF that ...
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64 views

A question about second-order logic and inaccessible cardinals.

Let $\kappa$ denote an inaccessible cardinal, and suppose $T \in V_\kappa$ is a second-order theory. Now consider some mathematical structure $X \in V_\kappa$. Then I think it is clear that $X \models ...
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56 views

Elementary equivalence of models

I'm quite new to model theory, so please correct me if I'm using wrong terminology. I need help with an exercise from Smirnov's book "Varieties of algebras" (In Russian). Problem: Assume that a ...
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1answer
38 views

Relative Interpretations alla Kunen

at the moment I try to figure out some details of Kunen's "Relative Interpretation" Definition (within the 2013 Edition of his "Set Theory", p. 99 to 100): Definition If $\Lambda$ is some axioms ...
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75 views

Decision and the Uncountable Spectrum

In 2000, Hart, Hrushovski, and Laskowski classified all complete first order theories in a countable language up to their uncountable spectra. However, does this also imply that given a $any$ ...
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1answer
73 views

Are algebraic structures required to satisfy axioms?

Is it a requirement for algebraic structures, when studying universal algebra, to satisfy axioms? The reason I ask is because algebraic structures are only defined by a underlying set, a signature, ...
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151 views

Elementary equivalence of free groups

This must be known inside out by model theorists by I have no cluse whether the following is true or not: Denote by $F_n$ the free group on $n$ generators. Suppose that $n\neq m$. Are the groups ...
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278 views

An Uncountable language , A Model of $\mathbb{N}$, A Problem.

Edit 1: I messed up my original question, but Arthur Fischer answered my question anyway. Edit 2: We can actually restrict $L$ to the language in arithmetic with the prdicate $P_{\mathbb{P}}$. ...
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301 views

Upper bound on cardinality of a field

Is there an upper bound on the cardinality of a field? The "biggest" fields I know are the field of real numbers, or the field of complex numbers. Is there a field with cardinality greater than ...
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90 views

Is infinitary logics $\mathcal{L}_{\infty\omega}$ an abstract logic?

Infinitary logics $\mathcal{L}_{\infty\omega}$ is an extension of first-order logics such that $\bigvee\Phi \in \mathcal{L}_{\infty\omega}$ if $\Phi$ is a set of ...
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1answer
31 views

Is “constructible from” a transitive relation?

In Jech's Set Theory, exercise 13.27, it is hinted that $X \in L[Y]$ and $Y \in L[X]$ together imply $L[X]=L[Y]$. I tried to prove this fact without success, although I suspect the proof is simple. ...
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164 views

Construction of Ultrafilters

I've been doing a lot of work with ultrapowers and saturation recently. In particular, I am reading chapter 6 of Chang and Keisler as well as Keisler's paper on "Ultraproducts which are not ...
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1answer
87 views

Is there an algebraic ω-categorical structure with quantifier eli., that is not ultrahomogenous?

The following result holds for relational structures: If $A$ is a countable structure, with an $\omega$−categorical theory $Th(A)$, that admits quantifier elimination, then $A$ is ultrahomogeneous. ...
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91 views

Why is $\alpha \mapsto L_{\alpha}[A]$ $\Delta_{1}$?

On page 187 of Jech's Set Theory, there is a proof sketch of the fact that $\alpha \mapsto L_{\alpha}$ is $\Delta_{1}$. As far as I can tell, Jech's argument only shows that this operation is ...
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44 views

Sets Constructible Relative To A Unary Predicate

The class $L$ of constructible sets is defined by recursion using the operation def$(M)=\{x \subset M: x$ is definable over $(M, \in) \}$. By adding a unary predicate, $P$, to our language, we can ...
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68 views

Model-theory : questions regarding back-and-forth sets

See my previous post for the basic definitions from Jouko Väänänen, Models and Games (2011), page 54-on. See page 64 for : Definition 5.14 Suppose $\mathcal A$ and $\mathcal B$ are ...
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117 views

Model-theory : questions regarding partial isomorphism

I'm having problems with the first pages of Bruno Poizat, A Course in Model: Theory An Introduction to Contemporary Mathematical Logic (ed or 1985), specifically with local isomorphism and back- and ...
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1answer
159 views

The connection between quantifier elimination, $\omega$-categorical and ultrahomogenous

A relational structure $A$ with an $\omega-$categorical theory $Th(A)$ is ultrahomogenous iff $Th(A)$ admits quantifier elimination. (*) I was wondering wether the structure $A$ has to be ...
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226 views

Gödel's way of teaching non-standard models to Takeuti.

In Memoirs of a Proof Theorist, Gaisi Takeuti relates how Gödel taught him about nonstandard models in an "interesting" way: It went as follows. Let T be a theory with a nonstandard model. By ...
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115 views

Is for every ultrahomogenous structure M the theory Th(M) model complete?

A structure M is ultrahomogenous if every isomorphism between finitely generated substructures of M can be extended to an automorphism of M. A theory is model complete if every embedding between ...
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85 views

Show that if $L$ is countable and contains a two-place predicate symbol, there are $2^{2^{\aleph_0}}$ classes of $L$-structures closed under $\equiv$

We say that a class of structures $K$ is closed under elementary equivalence ($\equiv$) if for all $A, B$, if $A \in K$ and $A \equiv B$, then $B \in K$. How to show that if $L$ (as a set of specific ...
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Model of complete extension of Zermelo set theory

Chang and Keisler's Model theory gives the following exercise problem: Prove that there is a complete extension $T$ of Zermelo set theory which has arbitrary large natural models. (A model ...
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64 views

Why are there $\leq$ $2^{\operatorname{card}(\operatorname{Form}(L))}$ elementarily nonequivalent structures for $L$?

Let $L$ be a set of specific symbols and $\operatorname{Form}(S)$ be the set of all first-order formulas over $L$. Why are there $\leq$ $2^{\operatorname{card}(\operatorname{Form}(L))}$ ...
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1answer
64 views

How to prove that $max(\aleph_{0}, card(X)) = max(\aleph_{0}, card(L(X)))$?

I struggle with the following problem. Let $X$ be a set of elementary sentences and $L(X)$ be the smallest elementary language in which we can express all the sentences from $X$. How to prove that ...
2
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1answer
120 views

Cardinalities of Collections of Models

Let $T$ be a complete theory in a countable language (with only infinite models). Recall the spectrum function: $I(\aleph_\alpha,T)=$ the number of non-isomorphic models of $T$ of cardinality ...
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75 views

A model which has only one undefinable element over a language with only a finite number of symbols

I try to solve the problem 1.3.14 in Chang and Keisler's Model theory: For each $n\in\omega$, find a model $\mathfrak{A}_n$ for $\mathcal{L}$ a language with only a finite number of symbols, which ...
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185 views

Can $(\Bbb N,\leq)$ have an $\aleph_0$-categorical theory (in a larger language)?

One of the nicer consequences of compactness is that there is no statement in first-order logic whose content "$\leq$ is a well-order". So we can show that there are countable structure $(M,\leq)$ ...
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Counter example of monotone union [duplicate]

I saw this exercise in "Elements of Abstract and Linear Algebra" by E. H. Connell: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, ...
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98 views

On the number of countable models of complete theories of models of ZFC [duplicate]

Fix the language of set theory $\mathcal{L}=\{\in\}$. Let $\langle M,\in\rangle$ be a set or proper class model of ZFC (e.g. $M$ could be $L$, $HOD$, $V_{\kappa}$ for some inaccessible cardinal ...
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1answer
149 views

Why is the cardinality of a language defined as $||\mathcal{L}||$?

I'm reading Chang and Keisler's Model Theory and I don't quite understand the notation they use for the cardinality of a language. Elsewhere in the book, the cardinality of a set $X$ is denoted by ...
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1answer
82 views

Inuition regarding Lowenheim-Skolem applied to models of set theory

According to wikipedia, ...the Löwenheim–Skolem Theorem states that for every signature $σ$, every infinite $σ$-structure $M$ and every infinite cardinal number $κ ≥ |σ|$, there is a ...
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86 views

One-element model of first-order PA

The First-Order axiomatisation of PA is: $\forall x. x = x$ $\forall x, y. x = y \rightarrow y = x$ $\forall x, y, z. x = y \land y = z \rightarrow x = z$ $\forall x. 0 \ne S(x)$ $\forall x, y. S(x) ...
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63 views

elementary class and abstract elementary class

I think I confused with the concept of elementary class and abstract elementary class. We see in the definition of AEC that each elementary class is an AEC. Let $l=\{\le\}$, $T=\emptyset$, ...
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Countable transitive $T \vDash ZFC-P$ with $\approx$ not absolute: why do we need $H(\aleph_3)$?

I am trying to solve Exercise IV.3.31 from Kunen's Foundations of Mathematics. I think I have a solution but I am confused by one of the hints. By request, here is the text of the exercise. ...
3
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1answer
63 views

Existence of theories with exactly two countable models

I read a result of Vaught(a little down the page) that says that there cannot be any first order theory which has exactly two countable models upto isomorphism. Is this not a counter example: The ...
2
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1answer
107 views

Absolute confusion! (A question about absolute *sentences*)

I'm seriously confused about absoluteness. A formula in the language of a theory $T$ is absolute for $T$ structures if its truth value is the same in all standard transitive models of $T$ (this may ...
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146 views

How large or small can the gap in Shelah's main gap theorem be, up to consistency?

Shelah's main gap theorem in model theory says: For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of ...
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3answers
64 views

The differences between Model Theory and Computation Theory

Model Theory seems more interested with Gödel's completeness theorem, Tarski' quantifier elimination and logic systems that Turing computability and Church recursivity. However, both theories overlap ...
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1answer
109 views

Elementary equivalence versus equivalence between the total theory in model theory

In the page for elementary equivalence on wikipedia, in the introduction, they say: "If N is a substructure of M, one often needs a stronger condition. In this case N is called an elementary ...
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55 views

Are there any purely semantic proofs of the compactness theorem that don't use the full axiom of choice? [duplicate]

Using Godel's completeness theorem, it can be shown that the compactness theorem is equivalent to the ultrafilter lemma. The compactness theorem can also be proven using ultraproducts and Los's ...
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Prove that ⊨ is not symmetric

Another question from a beginner: We want to prove that ⊨ is not symmetric by finidng concrete formulas φ and ψ for which we can show that φ ⊨ ψ and ψ ⊭ φ. Thank you for your help!
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Show that $(V_\omega,\in) \models \mathrm{Power\ set}$

Let "$\mathrm{Power\ set}$" denote the Power set axiom. I'd like if some of you could tell me if the solution of the following exercise is correct. I want to prove that $(V_\omega,\in)\models ...
3
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1answer
70 views

Is first-order logic complete with respect to countable structures?

Question. Let $\Gamma$ denote a first-order signature, and consider a sentence $\tau$ and a set of sentences $\Sigma$ in the language generated by $\Gamma$. If every countable $\Gamma$-structure that ...
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61 views

Nonstandard models of PA with a decidable order relation.

There this exercise in Models of Peano Arithmetic (Kaye 1991, p.157), which asks to define a recursive binary relation on $\mathbb{N}^2$, such that $M \upharpoonright < $ is isomorphic to ...
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62 views

Absolute formula

I'm trying to study the model of Set Theory and I have a problem. If I know that a formula is absolute for a class, could I infer that the formula is true in my class? Namely, I'm trying to prove in ...
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80 views

Examples of Jónsson Models

Let $T$ be a complete first order theory. Suppose that $M\models T$. Then, $M$ is said to be a Jónsson Model of $T$ if for all $N$, such that $N\prec M$ and $N\models T$, we have $|N|<|M|$ (Note ...