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 ...

learn more… | top users | synonyms

3
votes
1answer
31 views

Generalizations of pregeometries

Combinatorial geometries and pregeometries are important in classifying strongly minimal (as well as O-minimal) theories. More formally, a model of a strongly minimal (or an O-minimal) theory with the ...
0
votes
1answer
61 views

Hanf Numbers and Decidability

Currently reading J.L. Bell's Models and Ultraproducts and at the end of Chapter 4 section 4 the authors comment that "In spite of the fact that most languages can easily be shown to possess Hanf ...
1
vote
1answer
27 views

Prove that iff a formula $\phi (v_1, v_2,…v_n)$ is satisfied in the substructure $\mathcal M$, then it is satisfied in structure $\mathcal N$

Assume $\mathcal M \subseteq N$ structures for signature $S$. $\mathcal M$ is a substructure of $\mathcal N$. Let $\phi(v_1, \cdots v_n)$ be a formula without quantifiers. Prove by induction on ...
3
votes
1answer
53 views

Show that the set of all points $x \in \mathbb R$ where $f$ is differentiable is definable in $\mathcal M=(\mathbb R; +,-(), \cdot, \lt, 0,1,f)$

For the structure $\mathcal M=(\mathbb R; +,-(), \cdot, \lt, 0,1,f), n_f=1 $ show that the set of all points $x \in \mathbb R$ where $f$ is differentiable is a definable set. My issue here is how to ...
1
vote
2answers
36 views

Prove or disprove that for theory $T$, $T \vdash (\phi \rightarrow \psi) \iff T \cup \{\phi\} \vdash \psi$.

Prove or disprove that for theory $T$, $T \vdash (\phi \rightarrow \psi) \iff T \cup \{\phi\} \vdash \psi$. This seems quite right, but I don't know how to prove it. So lets start with ...
2
votes
2answers
56 views

Vacuously true? Prove or disprove that for every theory $T$, if $T$ is not satisfiable then for every $\phi$, $T \vdash \phi$

Is it vacuously true? Prove or disprove that for every theory $T$, if $T$ is not satisfiable then for every $\phi$, $T \vdash \phi$. If $T$ is not satisfiable, then there is no structure ...
0
votes
1answer
46 views

A singleton set $\{g\}$ can be regarded as a unary relation in $G$. Why?

Theorem 1.1. A relation $R \subseteq M^n$ is definable if and only if every automorphism of every elementary extension of $M$ preserves $R$. For a proof, the reader can see [4]. Suppose we ...
3
votes
2answers
34 views

Functions or relations stable under automorphism

Suppose we have a structure $M$, that is, a set $S$ with some designated functions and/or relations on that set. We can define automorphisms for this structure. What is the term in the standard logic ...
0
votes
1answer
19 views

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 ...
1
vote
1answer
42 views

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 ...
2
votes
3answers
143 views

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?
0
votes
0answers
23 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 ...
2
votes
2answers
411 views

What is the real meaning of Hilbert's axiom of completeness

According to Greenberg's book of geometry it is sufficient to consider the axiom of Dedekind along with Hilbert's axioms (except of course for the Archimedian Principle and his Axiom of Completeness) ...
4
votes
1answer
81 views

Models of infinite groups and 'Group-like' objects

Let $G$ be an infinite group, and for simplicity, we will assume that $G$ is also countable. Now, with $G$ in mind, we construct a new language $L_G=\{f_{a_i\_},f_{\_a_i}:a_i\in G\}$ where ...
5
votes
3answers
93 views

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. ...
2
votes
0answers
79 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 ...
3
votes
0answers
52 views

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/ ...
4
votes
1answer
61 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 ...
0
votes
1answer
50 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 $\mathfrak{A}_i \vDash \varphi$ ...
10
votes
2answers
312 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 ...
1
vote
1answer
47 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 ...
6
votes
1answer
300 views

Difference between elementary submodel and elementary substructure

This is a really "elementary" question, forgive the pun. What is the difference between an elementary submodel and an elementary substructure (in first-order Logic)? Sincere thanks for help.
2
votes
1answer
40 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 ...
3
votes
1answer
40 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 ...
6
votes
0answers
75 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 $r\in\mathbb{N}$, ...
7
votes
0answers
224 views

Gödel's Completeness Theorem and logical consequence

At the end of a long process of "rumination" on "old" math log textbooks, I've found the "missing link" - from my personal point of view - between some issues I've raised in the previous months : (i) ...
1
vote
2answers
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) ...
1
vote
1answer
71 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 ...
1
vote
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}) ...
6
votes
1answer
364 views

Structuralist slogans

I am afraid to make a bad impression by misusing this forum but I am looking for as-many-as-possible mathematically inspired formulations and references to one (sometimes vague) idea. The idea is ...
3
votes
1answer
75 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) ...
5
votes
0answers
50 views

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 ...
6
votes
6answers
440 views

In what sense of “structure” do group homomorphisms “preserve structure”?

It is commonly said that group homomorphisms "preserve the structure of the group", e.g., from Wikipedia: The purpose of defining a group homomorphism as it is, is to create functions that ...
2
votes
0answers
38 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 ...
1
vote
1answer
31 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 ...
1
vote
1answer
69 views

About the proof of a test for quantifier elimination.

I've been reading D. Marker's book on Model theory. In the part dealing with quantifier elimination there's a corollary I've been trying to prove without any luck: Corollary 3.1.6 Let $T$ be an ...
2
votes
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 ...
4
votes
2answers
83 views

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 ...
1
vote
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 ...
1
vote
1answer
28 views

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 ...
1
vote
2answers
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 ...
1
vote
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 ...
4
votes
2answers
221 views

Characterization of totally categorical theories

I have what I am sure is a trivial question, but I can't seem to answer it for myself. In model theory, there is a theorem of Hrushovski which shows that if T is a totally categorical theory (i.e., T ...
21
votes
1answer
573 views
1
vote
0answers
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 ...
-2
votes
0answers
43 views

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 ...
2
votes
1answer
67 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 ...
0
votes
2answers
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: $K_1=Ass(X)=\{ v|v \vDash X \}$ ...
1
vote
0answers
32 views

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 ...
0
votes
0answers
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 ...