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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Quickie on Boolean valued models

Bell writes on page 21 (you may use the search in the preview to search for "21" to view the page): "..., we show that, for any complete Boolean algebra $B$, all the theorems of $ZFC$ are true in ...
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Question about passage in Halbeisen's book

I am looking at the following passage in Halbeisen's book "Combinatorial Set Theory" (p 260 at the bottom): What is the role of $\Phi$? It seems to me that a finite fragment is the same as a ...
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130 views

Question about the proof of $GCH$ holds in $\mathbf L$

I have a question about the proof of the following: (Lemma 37) Assume $\mathbf V = \mathbf L$, and let $\kappa$ be a cardinal. Then $\mathcal P (\kappa ) \subseteq L_{\kappa^+}$. Assume we ...
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Applications of model theory to analysis

Some of the more organic theories considered in model theory (other than set theory, which, from what I've seen, seems to be quite distinct from "mainstream" model theory) are those which arise from ...
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Example of a set that is in $\mathbf V$ but not in $\mathbf L$

Let $\mathbf V$ denote the cumulative hierarchy and let $\mathbf L$ denote Gödel's constructible universe. We then have $\mathbf L \subseteq \mathbf V$. Would someone give me an example of a set that ...
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60 views

Possible typo in Just/Weese's set theory

In Just Weese on page 197 there are the following corollaries: Regarding Corollary 24: Is this a typo and should say "$CON(ZF) \not\rightarrow CON(ZF + \exists \text{ "a strongly ...
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145 views

Understanding the model-theoretic proof of Hilbert's Nullstellensatz

The proof I am talking about goes like this: Given $k$ algebraically closed and $(f_1,..,f_k)=I\neq (1)$ an ideal in $A=k[x_1,..,x_n]$, let $m$ be a maximal ideal with $I\subseteq m$ and observe that ...
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143 views

Proving Axiom Schema of Replacement holds in $H_\lambda$

I think I proved the following, can you tell me if my proof is correct? Exercise 23: Show that if $\lambda$ is a regular infinite cardinal then $\langle H_\lambda , \overline{\in} \rangle$ satisfies ...
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249 views

How to break power set for non-transitive models?

Apparently, $\varphi (x,y) : y = P(x)$ where $P$ denotes the power set is not absolute for transitive models. We call a formula $\varphi(v_1, \dots v_n)$ in $L_S$ absolute for a class $\mathbf X$ ...
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407 views

Model Theory and Topology Connections

I have studied a bit of model theory, when I say "a bit" I have studied much more than is available to a typical undergraduate in the UK (i think, certainly from what I have seen) but I am sure this ...
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130 views

Extensionality of a relation and Axiom of Extensionality

What's the difference between the Axiom of Extensionality $(A1)$ and an extensional relation? The definitions are $(A1) \forall x,y ( x = y \leftrightarrow \forall z ( z \in x \leftrightarrow z \in ...
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167 views

A question about standard models

As I understand it a standard model is a model in which the relation is the $\in$ on the actual set of sets constituting the model. (i) Hence theories that aren't in the language of set $L_S$ ...
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176 views

A question about a passage in Just/Weese's Basic Set Theory

I have a question regarding the following passage in Just/Weese (p 192), $\mathbf{V}$ denoting the cumulative hierarchy: "But consider the following situation: Where in "($\beta$)" does the ...
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119 views

Question about cumulative hierarchy

In the following let $\mathbf{V} = \bigcup_{\alpha \in \mathbf{ON}} V_\alpha$ denote the cumulative hierarchy. Let $\{\varphi_0, \dots, \varphi_n, \dots \}$ denote a list of all $ZF$ axioms. I am ...
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200 views

Can we prove the completeness of FOL based on forcing?

In David Marker's Model Theory, there is a exercise said " we can view a countable Henkin construction as a forcing construction ". But in the book the Henkin construction is used to prove the ...
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65 views

Model complete theories of henselian local rings which are not nec valuation rings

I just want to ask if anybody as any examples of a first order model complete theorie of henselian local rings which is not some theory of valuation rings. More precisely- I am looking for a theory ...
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333 views

Does quantifier elimination imply decidability?

I know in some simple cases of quantifier elimination that I have seen, one ends up seeing that the process of quantifier elimination resulted in being able to show decidability. Is this true in ...
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205 views

Is every model of PA well-ordered?

Assume first-order Peano arithmetic is consistent and $N$ is its model, we know that every subset of $N$ contains a minimal element. It's a second-order property so I am not sure if it hold in ...
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587 views

Does every complete theory admit quantifier elimination?

Does every complete theory admit quantifier elimination? I know that at least in some simple cases the reverse is true; such as some reducts of number theory.Thanks
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94 views

Show that the theory of the $Th(\mathbb N)$ in first order logic with the finiteness quantifier is categorical

Suppose that a finiteness quantifier $\mathbf Fx$ is added to first order logic. Its semantics are: $\mathbf Fx\Phi(x)$ is true in a model just in case there a finitely many things in the domain of ...
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207 views

$\langle \Bbb Q,<\rangle$ is an elementary submodel of $ \langle \Bbb R,< \rangle$

I am trying to show that $\langle\Bbb Q,<\rangle$ is an elementary submodel of $\langle\Bbb R,<\rangle$. I first believed that this problem is quite trivial $-$ I thought all I needed to do was ...
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145 views

Automorphism of an elementary extension of a structure that moves an undefinable element

I know that the easiest way to show a point is not definable is to find an automorphism of the structure that moves the given point. I've also seen many examples undefinable points that couldn't be ...
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230 views

Types and Algebraicity

Let $\Phi(\bar x)$ be a type over a set $X$ with respect to a structure $A$. Show that if $\Phi$ is algebraic, then $\Phi$ contains a formula $\phi$ s.t. $A\models\exists\ _{<n}\bar x\phi(\bar x)$ ...
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241 views

Consequences of axioms of Dense linear order without endpoints is complete

I've been musing over this problem over the past few days, and believe I have an answer. However, I am still a bit shaky with some of the definitions I am using, and would appreciate if anyone could ...
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Model existence for infinitary logics

One of the problems of infinitary logic is that it is possible for compactness to fail in a spectacular way: for example, one can concoct an inconsistent set of axioms whose proper subsets are all ...
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Definability in a given structure

I want to prove the following statements: Is the function "sin" definable in the structure $(\Bbb{R},<,+,\cdot,0,1)$, that is does there exists a formula $\phi=\phi(x_0,x_1)$ such that for all ...
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222 views

Does elementary embedding exist between two elementary equivalent structures?

By previous question, if there is a elementary embedding from $\mathfrak A$ into $\mathfrak B$, then $\mathfrak A \equiv \mathfrak B$. Now it is naturally to ask conversely, if $\mathfrak A \equiv ...
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351 views

Is one structure elementary equivalent to its elementary extension?

Let $\mathfrak A,\mathfrak A^*$ be $\mathcal L$-structures and $\mathfrak A \preceq \mathfrak A^*$. That implies forall n-ary formula $\varphi(\bar{v})$ in $\mathcal L$ and $\bar{a} \in \mathfrak A^n$ ...
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313 views

Quantifier Elimination

I want to prove the following: The structure $(\mathbb{Z},\equiv,0)$ has QE (with $\equiv$ a relation such that for all $m,n\in\mathbb{Z}$: $m\equiv n$ iff $m-n$ is even). I thought hereover in the ...
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384 views

Automorphisms of saturated models

This is basically Exercise 10.1.5(c) in Hodges's Model theory. First, a reminder of some definitions: Let $\lambda$ be a cardinal, and let $\Sigma$ be a finitary first-order signature. A ...
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142 views

Feferman-Vaught theorem and Term Powers

In an example of usage of quantifier elimination in wikipedia, it briefly mentions Feferman-Vaught theorem and Term Powers, but I am finding little information on what these are. Can anyone explain ...
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229 views

Is Zermelo set theory finitely axiomatizable?

I know that ZF is not finitely axiomatizable, but what about Z (i.e. ZF without Replacement)?
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A formula that is upwards absolute but not downwards absolute

I know that any existential formula is upwards absolute (an any universal formula is downwards absolute) but I was looking for an example of a formula that is upwards absolute but not downwards ...
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Definable relations

I study model theory and I have questions about relations which are definable in a structure or not. I found three examples from exercises and i want to do them: Is the relation $<$ on $\Bbb{Q}$ ...
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153 views

Is $\mathbb N$ definable in $\mathbb C$?

$\mathbb C$ is an algebraic closed field with characteristic $0$, hence $Th(\mathbb C)$ is a recursive satisfiable complete theory, thus recursive axiomatizable. So if $\mathbb N$ is definable in ...
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Is this theory $\kappa$-categorical when $\kappa$ is infinite and not $\le \aleph_0$?

Let $\mathcal L$ be a language with only a binary predicate $E$, and let $T$ be a theory of structures in which $E$ is an equivalence relation which partitions the structure into two infinite ...
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262 views

Showing that the class of well ordered structures is not an elementary class

How I can prove that for the class $K$ of well-ordered structures there is no finite set of statements $T$ such that $\text{Mod} (T) = K$? Thanks
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219 views

Is the negation of Gödel's completeness theorem consistent with $ZF$ without AC?

The proof of compactness and completeness of $\mathscr{FOL}$ (with Hilbert system) used Zorn's lemma. And Zorn's lemma is equivalent to the Axiom of choice in $ZF$. So my question is can they be ...
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194 views

Finding the exactly number of countable models of a theory

I'm working for Modeltheory and i have the follwoing information: Define $T_0=Th((\mathbb{Q},<,0,1,2,\cdots))$ and ...
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53 views

Does one restrict oneself when leaving out the inversion in the signature of groups?

I feel a bit lazy, for I don't want to seek an answer myself in some textbook – because I fear it'd cost me too much time while someone here can easily just knock out the answer. I hope that's OK. (Do ...
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113 views

Proving that $\langle \omega, \in \rangle $ models empty set and extensionality

Let (A1) be the axiom of extensionality: $\forall x,y ( x = y \longleftrightarrow \forall z \in x \leftrightarrow z \in y))$ and let (A2) be the empty set axiom $\exists x \forall y (y \notin x)$. ...
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44 views

Model of a language $L$ vs. model of a theory $T$ in $L$

I am reading Just/Weese and they seem to use "model of a language $L$", for example, p. 90: and, more disturbingly, p. 91: Isn't this a "typo" (or perhaps sloppy writing)? If $L$ is any language ...
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51 views

Question about a proof that there is no formula $fin(x)$

Consider the following passage in Just/Weese: where $fin(x) \equiv $ "$x$ is a finite set". I'd like to confirm the following with you : requiring "ZFC $\vdash fin(n)$ if $n \in \omega$" is weaker ...
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70 views

Borel sets used in Model Theory

In Lemma 4.4.13 (page 161) of D.Marker- Model Theory book it is proved that $D(F,T)$ (the set of all possible F-diagrams of models of T) is a Borel set. Can someone explain to me the proof given a ...
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A theory $T$ is complete if and only if $Th(M) = T^\vdash$ for some model $M$

I'm trying to show that a theory $T$ is complete if and only if $Th(M) = T^\vdash$ for some model $M$ where $Th(M)$ denotes the set of all sentences that are true in $M$. What I have so far: ...
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Does $E$ in a model $\langle M, E\rangle$ of ZFC have to be wellfounded?

Consider the following exercise from Just/Weese: My first reaction was "Of course $E$ has to be strictly wellfounded otherwise it wouldn't model $\in$" but apparently I am missing something since I ...
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Confused about models of ZFC and passage of book

I have a question about a passage of Just/Weese. In the following, let $\mathcal M = \langle M, E \rangle$ be a model of ZFC. Here is the first half of what I'm about to ask a question: So I asked ...
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122 views

How to finish proof that $T$ has an infinite model?

I'm trying to prove the following: If $T$ is a first-order theory with the property that for every natural number $n$ there is a natural number $m>n$ such that $T$ has an $m$-element model then $T$ ...
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53 views

Equivalent characterisation of consistent

We call a set of formulas $\Sigma$ of a language $L$ consistent if there is no $\varphi$ in $L$ such that $\Sigma \vdash \varphi$ and $\Sigma \vdash \lnot \varphi$. Apparently, an equivalent ...
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Choosing a Master Thesis Topic: Logic - Model Theory

I am a first-year graduate student in maths. Around these days, I feel I must decide on which exact part of mathematics I shall go through. Infact, I have narrowed down the suitable options but still ...