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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Saturated models and $\kappa=\kappa^{<\kappa}$

Do not assume GCH. Can you characterize the cardinals $\kappa$ such that every theory $T$ with an infinite model has a saturated model of cardinality $\kappa$? I guess these are the cardinals such ...
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Ordered field of rationals axiomatizable

Is there a set of sentences in the language of ordered fields whose models are precisely the rationals and any ordered field isomorphic to them?
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Reflexive reduct of preorder

Suppose P is a preorder on a set S, a reflexive and transitive relation. Suppose we subtract from P the identity relation and get a relation Q on S. Is the class of all such relations a first-order ...
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20 views

Definable valuation ring

If $(K,v)$ is henselian and $O_v$ is $\phi$-$\text{definable}$, why do I have that if $L\equiv K$ (in the language of ring) then $L$ admits a non-trivial henselian valuation ring? I understand if ...
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Complete extensions of a consistent theory

I understand that I need to use compactness but somehow can't finish it. Suppose $L$ is a language and $T$ a consistent $L$-theory with only finitely many logically inequivalent complete extensions. ...
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Is every reduced $k$-algebra all of whose residue fields are $k$ finitely generated?

Let $k$ be a field (of characteristic zero if you want). Let $A$ be a reduced $k$-algebra with the property that for every prime ideal $\mathfrak{p}$ of $A$ the natural homomorphism $k \to A/ ...
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78 views

Relative Identity vs Set Theory

The last complete, but unpublished, paper by the late Tom Etter titled "Three-Place Identity" purports to prove that all of mathematics can be expressed in terms of relative identity. In his own ...
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49 views

show that a horn sentence is preserved under a direct product.

show that a horn sentence is preserved under a direct product. If $\varphi$ is a horn sentence and $\mathfrak{A}_i, i \in \text{I}$ is a model for $\varphi$ namely $\mathfrak{A}_i \vDash \varphi$ ...
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46 views

Indiscernibles and colorations.

Let $L$ be a language, and $(X,\leq)$ be a total order contained in an $L$-structure $\frak{A}$. Now if we denote by $[X]^n$ the set of $n$-sized sequences in $X$ and consider a set $\Gamma$ of ...
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296 views

Is algebra needed to really understand and/or enjoy model theory?

What are the desirable pre-requisites to be able to learn model theory well? In particular, it seems that connections to algebra are used heavily especially as examples. I would like to know if a ...
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39 views

Proving that every submodel of DLOE is an elementary submodel

Let $A$ and $B$ be models of the theory of Dense Linear Orders without Endpoints such that $|B| \subset |A|$. I'm trying to prove that $B$ is an elementary submodel of $A$. Using Tarski-Vaught test I ...
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39 views

Homogeneous models in a strongly minimal theory

I am trying to prove that every infinite model of a complete strongly minimal theory T is homogeneous. Clearly, if $k:M \to M $ is a partial elementary map with $|k|<|M|$ and $M$ is a model of ...
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27 views

How Many Countable Models of the successor function

Consider the successor function (s(x)=x+1), $T_{S}$ to be the set of axioms given by; S1: ∀xy[s(x)=s(y)→x=y] (injective) S2: [s(x)≠0] (never 0) S3: ∀x[x≠0→∃y[s(y)=x]] (everything bar 0 is in image) ...
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70 views

On proving the zero-one-law for first order logic

I'm trying to understand the proof of the zero-one-law for first order logic as provided in (Ebbinghaus-Flum, 1995). It goes as follows: Let $\tau$ be a relational signature. Let $r\in\mathbb{N}$, ...
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1answer
27 views

Construction of an elementary extension satisfying $Th(\mathcal{M}) \cup S$ where $S(x)$ is an arbitrary set of formulas

Let $S(x)$ be a set of $\mathcal{L}$-formulas (containing at most the free variable $x$). Is there an elementary extension $\mathcal{N}$ of $\mathcal{M}$ such that $\mathcal{N} \models Th(\mathcal{M}) ...
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Deductive closure of sentence $\forall x \forall y F(x,y) \stackrel{.}{=} F(y,x)$ in language $\mathcal{L}$ is undecidable.

$\mathcal{L}$ is the language that contains a single binary function symbol $F$. In the earlier parts of this question, we were told to take the $\mathcal{L}$-structure $\mathcal{M}$ with universe ...
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1answer
69 views

Non-standard model of arithmetic and Gödel's theorem [closed]

This is a cross-post of a question asked on History of Science and Mathematics Stack Exchange. I've read Skolem's paper on his non-standard models of the arithmetic ("Über die ...
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36 views

Models and signatures for propositional logic

The following is a bit long, so I collected my questions at the end, but as this is the only opportunity I get for feedback I would appreciate it if anyone could also point out where I've gone astray ...
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57 views

What is the degree of a real closure of an ordered field?

Given a subfield $F$ of $\Bbb R$, let the real-closure of $F$ be the smallest subfield of $\Bbb R$ which is real-closed and extends $F$. Since the theory of real-closed fields is complete, this is in ...
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68 views

Models of the successor function

I would like to ask a few questions about models of the succesor function (s(x)=x+1), intact that is a bit vague, consider $T_{S}$ to be the set of axioms given by; S1: $\forall xy[s(x)=s(y) ...
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1answer
30 views

Axiomatizability in monadic second-order logic

For my thesis in finite model theory I'm considering some basic classes of structures, and I want to show in which logical systems they can or cannot be axiomatized. I now consider the class ...
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Every theory eliminates quantifiers in an appropriate definitional expansion?

I need to prove that every theory eliminates quantifiers in an appropriate definitional expansion. For this, consider: let $T$ be a theory in language $L$. Consider the following expansion of the ...
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1answer
15 views

Characterization of superstability

In a stable theory every global type $p$ is invariant (= non-forking) over ${\rm acl^{eq}}(A)$ for some set $A$. Is there a characterization of superstability and/or $\omega$-stability in terms of the ...
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If for any $M' \subseteq M$ there is an embedding of $M'$ into a $Mod(T)$, then there is an embedding of $M$ into $Mod(T)$.

I need to prove that, for $M$ a given $L$-structure and $T$ be a theory in the language $L$. Show that if for any finitely generated substructure $M'$ of $M$ there is an embedding of $M'$ into a model ...
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64 views

Some exercises on Models in First Order Logic [closed]

So these are some practice exercises for a Math Logic exam, I can't get a hold on how to do these. Semantically what does T = Th(N) represent? And how do I go about constructing a model A for T, ...
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35 views

Non isolated types of $\mathcal M$ cannot be isolated in $\mathcal N \succ \mathcal M$?

Suppose $a \in M$ realizes a non-isolated type over $\emptyset$, and let $\mathcal N \succ \mathcal M$, furthermore let $|\mathcal M| = \aleph_o$ while $|\mathcal N| = \aleph_1$. Is it true that the ...
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1answer
46 views

What is the stone space $S_n(T)$ for a theory with infinitely many equivalence classes, each class infinite?

Let $L$ be the (first-order) language with one binary relation symbol $E$, and $T$ be the $L$-theory asserting that $E$ is an equivalence relation with infinitely many classes, each of which is ...
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28 views

Proof verification for structure construction

This question is from Enderton's mathematical logic. Question 8 sec 2.5 pg 146. It says assume the language that has $\forall$ and P, where P is a two place predicate symbol. Let A be the structure ...
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1answer
71 views

Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2

While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see [1]), these axioms seem somewhat arbitrary (e.g. adding an axiom that ...
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K is finitely definable if it has a finite support

I tried to prove that, but without a succes: Prove that K is finitely definable if and only if it has a finite support. *support of a set of assignments K is a set S that contains the atomic ...
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Union of definable sets is a definable set [duplicate]

I tried to prove this question but without a success: Let $K_1 \text{and } K_2$ be definable sets, prove that $K_1∪K_2$ is definable. What I tried to do is to assume: $K_1=\text{Ass}(X)=\{v\mid ...
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1answer
65 views

Exercise $ 3.4.15 $ of David Marker’s “Model Theory”.

I was reading David Marker’s Model Theory and came upon the following problem in Chapter 3. Setting Let $ \mathcal{M} $ be a saturated $ \mathcal{L} $-structure. A definable subset $ X \subseteq M ...
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127 views

Union of definable sets

I tried to prove this question but without a success: Let $K_1$ and $K_2$ be definable sets, prove that $K_1\cup K_2$ is definable. What I tried to do is to assume: $K_1=Ass(X)=\{ v|v \vDash X \}$ ...
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35 views

Examples and applications of homogeneus models in model theory.

Does anyone know any specific examples or applications of homogeneus models, to model theory or any other branch? For example, an application would be that prime models are isomorphic in a countable ...
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1answer
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How can the axioms (and primitives) of Tarski's axiomatization of $\Bbb R$ be independent?

While reading through this Wikipedia page about Tarski's axiomatization of the reals, a particular bit of text jumped out at me: Tarski proved these 8 axioms and 4 primitive notions independent. ...
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1answer
33 views

Sequence of indiscernibles in a theory with an equivalence relation with infinitely many equivalence classes

Let $\mathcal L$ be a language with a single binary relation $E$, and the theory $T$ where $E$ an equivalence relation with infinitely many equivalence classes, each of which is infinite. Are its ...
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1answer
45 views

Show that if $T$ is not $\aleph_0$-categorical then $T$ has a non-atomic model of size $\aleph_1$

Exactly as the title stated: Show that if $T$ is not $\aleph_0$-categorical then $T$ has a non-atomic model of size $\aleph_1$ Would like some pointers on how to proceed.
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1answer
51 views

What is satisfiable by a reduct of a model is satisfiable by the original model (and vice versa)?

My professor told me that any formula that is satisfiable by a reduct of a model is satisfiable by the model it is a reduct of, and vice versa (as long as the formula is interpretable on the ...
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1answer
46 views

What is the theorem that shows that second-order logic is the ceiling of model characterization?

I was reading this blog posting and the following claim was made: ...[T]here's tricks for making second-order logic encode any proposition in third-order logic and so on. If there's a collection ...
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Category of $\mathcal{L}$-structures

Given a purely relational language $\mathcal{L}$, the set of $\mathcal{L}$-structures forms a category under the definition of homomorphism found here. Call this category ...
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3answers
56 views

Proving uncountability of $\mathbb R$ only using the complete ordered field axioms

If we define the real numbers abstractly as a complete ordered field (like described in the Wikipedia page), how can we prove that they are uncountable? In other words, using just the axioms of a ...
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Morley’s Categoricity Theorem for uncountable languages.

Where can I find an accessible exposition of Shelah’s generalization of Morley’s theorem to uncountable languages? (Please, do not answer “Shelah’s Classification Theory”.)
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Why are structures with no relations called algebras?

"If [a given structure] A has no relations it is termed an algebraic structure, or simply an algebra" - Wolfgang Rautenberg, A Concise Introduction to Mathematical Logic, 3rd edition, page 42. I ...
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Show For any language L two L-structures M and N are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage.

Setting For any language $\mathcal L$, two $\mathcal L$-structures $\mathcal M$ and $\mathcal N$ are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage. ...
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Comparing Category Theory and Model Theory for Master's Thesis.

I am currently doing a Masters thesis in pure maths, and the two current fields that excite me are Category Theory (CT) and Model Theory (MT). I have been reading up on David Marker's Model Theory: ...
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1answer
42 views

Are the implicitly definable sets of a second-order theory the sets the second-order quantifiers range over?

I know that in a second-order setting, due to the failure of the Beth definability theorem, implicit and explicit definition come apart (i.e., there are predicates which can be implicitly, but not ...
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Given L = {<,c0,c1,…} and T3 the theory of DLO with sentence asserting co < c1 < …, Show T4 is complete with four countable models.

Let $\mathcal L_3 = \{<,c_o,c_1,\ldots\}$, and $T_3$ be the theory of DLO with sentences added stating $c_o < c_1 < \ldots$. Now let $\mathcal L_4 = \mathcal L_3 \cup \{P\}$, where $P$ is a ...
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1answer
40 views

Link between definitional expansions and definitional extensions.

I need to prove this, Let $T$ be a theory in language $L$, let $T'$ be a definitional extension of $T$ to language $L\subseteq L'$. If $\mathcal {M} \models T'$, then $\mathcal M$ is a ...
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62 views

What are the L-sentences that are true in an empty structure?

I am looking for an algorithm or set of rules to figure out whether a sentence (in first order logic) is true when we are dealing with an empty set as domain. Clearly, it has to be a sentence (no free ...
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(stability-theoretic) ¨weakly normal groups" are closed under subgroups

Let me first introduce two definitions: For a structure $\mathcal{M}$ in a language $\mathscr{L}$ and a subset $X \subseteq M^n$, the fully induced structure on $X$ is a structure $\mathcal{X}$ with ...