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

0
votes
0answers
16 views

Nonaxiomatizability one-dimensional vector spaces [duplicate]

Prove, class of all one-dimensional vector spaces over R isn't axiomatizable in the signature σ=< +, α, 0>, α* is single function vector multiplication by a scalar α of R.
1
vote
1answer
83 views

Why is class of one-dimensional vector spaces not axiomatizable?

Prove, class of all one-dimensional vector spaces over R isn't axiomatizable in the signature σ=< +, α, 0>, α* is single function vector multiplication by a scalar $\alpha$ of R. There are ...
2
votes
3answers
40 views

truth of a sentence to the linearly ordered set

Let Φ - sentence of signature σ = <≤> such that for any infinite linearly ordered set A satisfies A ⊨ F. Prove that there exists n ∈ N such that for every linearly ordered set B power greater than ...
2
votes
0answers
70 views

Lowering the power of infinite model

I need to prove that for every infinite model $\mathfrak A$ of signature $\sigma$ exists model $\mathfrak B$ with attributes: $\mathfrak A \equiv \mathfrak B$. $\parallel \mathfrak B \parallel = ...
0
votes
1answer
64 views

Order type of standard models of arithmetic

The standard model of PA has order type $\omega$. By compactness PA has a model of order type $\omega+n$ for any $n$, since every finite subset of the following set of statements is provable: ...
1
vote
0answers
81 views

Building sequence of axiomatizable classes that…

Any tips how to build sequence of axiomatizable classes $\mathrm{K_0, K_1, ..., K_n, ...}$ that class $\bigcup_{n\in w}\mathrm{K_n}$ is not axiomatizable?
3
votes
1answer
77 views

Elementary Submodels in Set Theory

I was reading the following summary on elementary submodels: http://boolesrings.org/mpawliuk/2012/01/26/a-practical-guide-to-using-countable-elementary-submodels/ Say $M\prec N$. The link above ...
2
votes
2answers
144 views

For types, is being definable is strictly stronger than being isolated?

Let $A$ be an $L$-structure and let $X\subset A$ and $a\in A$ be a tuple, and consider the type $p=\mbox{tp}_A(a/X)$. Let $T$ be the theory of $A$. Definition 1: $p$ is said to be isolated if for ...
2
votes
1answer
85 views

Some theorems of model theory

Do theorems like "omitting types theorem", "Extended completeness theorem" etc.. hold inside arbitrary countable transitive models of ZFC?
4
votes
1answer
54 views

Space of countable models of a theory T as a Polish space

Someone told me recently that the space of countable models of a first order theory $T$ form a Polish space. Can some one describe this construction to me?
0
votes
1answer
69 views

Is every model of $\Gamma$ a model of $Cn(\Gamma)$?

Is every model of $\Gamma$ a model of $Cn(\Gamma)$ ? $Cn(\Gamma)=\{\sigma:\Gamma \models \sigma\}$ This is the set of all sentences logically implied by $\Gamma$ . This could help me to understand ...
3
votes
1answer
61 views

First order logic: intersection is infinite

I am trying to solve my friend's homework assignment, I got stuck at this part: Let $\mathcal{L} = \{P^1, P^2, P^3, \cdots\}$ be language with equality, where $P^i$'s are unary predicates (relation ...
2
votes
1answer
57 views

Possible Turing degrees of countable models of ZFC

Let $M$ be a countable model in a signature $\Sigma$. We assume $\Sigma$ is finite, and (for convenience) has no function or constant symbols. Without loss of generality, we can assume that $M$'s ...
2
votes
2answers
100 views

Ultrafilter problem [duplicate]

could you help me with this problem, please? If $U$ is a principal ultrafilter on $I$ such that $\{a\}\in U$. Show that $Ult(\mathfrak{A}_x:x\in I)$ is isomorphic to $\mathfrak{A}_a$ and $[f]=f(a)$ ...
-1
votes
1answer
120 views

A finite subset of a countable, $\aleph_0$-categorical model invariant under automorphsims is definable? [closed]

sorry to bother you, but I got another question. I appreciate all your comments. Thanks a lot. Let $\mathfrak{A}$ be countable and ${\aleph}_0$-categorical. If $X \subset |\mathfrak{A}|^n$ is ...
1
vote
2answers
135 views

Model Theory problem

Good afternoon. I need some help with this little problem. I hope somebody could help me. Thanks a lot Assume that $A\equiv B$. Then there exists a $C$ such that $A\prec C$ and $B\prec C$.
2
votes
1answer
114 views

Show that if U is a principal ultrafilter then the canonical inmersion is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$

Show that: If $U$ is a principal ultrafilter, then the canonical inmersion $j$ is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$
1
vote
1answer
111 views

Show that $\Gamma$ is $\kappa $-categorical for $\kappa>\aleph_{0}$ , but not $\aleph_{0}$-categorical.

Let $\mathcal L=\{c_{i}:i<\omega\}$ be a language in first order logic, and: $\Gamma =\{\forall x(x=x),\forall x\forall y(x=y\rightarrow y=x),\forall x\forall y\forall z(x=y\wedge y=z\rightarrow ...
2
votes
1answer
111 views

Set of formulas in Model Theory

I'm reading the book Model Theory by Chang and Keisler and there is one thing that always bugs me. Very frequently we have something like $\Sigma(x)$ representing the set of all formulas in a language ...
3
votes
1answer
77 views

Model theoretic answer for having algebraic closure

I am beginner at the model theory and I learn compactness theorem at the class and I saw some application of it and one of them is that "every field has an algebraic closure". How can I prove it with ...
2
votes
0answers
105 views

Recursive non-standard models?

Any algebraically closed field (ACF) is a model of Modular Arithmetic (MA). MA has the same axioms as first order Peano arithmetic except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. MA ...
0
votes
2answers
106 views

Compactness Theorem explanation

Compactness Theorem definition: If $T$ is a theory in a first-order language $L$, then $T$ has a model iff every finite subset $S$ of $T$ has a model. A number of questions regarding to this ...
3
votes
1answer
39 views

Why do we need sometimes other structures than mentioned in the theorem to prove theorems?

For example when one proves that the elementary theory of finite fields is decidable, one uses pseudo-finite fields which are not in generally finite fields. Why do we need such a larger fields to ...
0
votes
1answer
47 views

2-type not-realised in Q

my question is the following: given the additive group of rational numbers, i.e. $Q = \langle {\mathbb Q},+,0\rangle$ and $T$ the theory of $Q$, how can I find (explicitly) a 2-type which is not ...
3
votes
1answer
67 views

Finitely many countable models implies decidability

Suppose $T$ is decidably axiomatizable first order theory and has no finite model. We shall focus on countable models. If $T$ has just one countable model (up to isomorphism), which means $T$ is ...
2
votes
1answer
68 views

Can axioms of the Euclidean space be proven in the Real space?

I'm not a mathematical logic student so my question can be naïve. I'm thinking if we identify $\Bbb{E}^2$ with $\Bbb{R}^2$( as a normed space with the Euclidean norm), Then can we prove all Euclidean ...
0
votes
1answer
96 views

A Question Regarding Uncountable Standard Models of ZFC Where CH is False

Let M be an uncountable standard model of ZFC, let $\frak c$ be the cardinality of the continuum, and let (just for the sake of argument) $\mathfrak c=\aleph_2$. If one assumes M has 'all' the ...
2
votes
1answer
67 views

Showing a Theory $T$ is Substructure Complete

Let $T$ be a (complete and consistent) theory. Suppose $T$ exhibits the following two properties: (1) model-completeness: if $\mathcal{M} \models T$ and $\mathcal{A} \subseteq \mathcal{M}$ s.t. ...
1
vote
1answer
40 views

Introduction to Valued Fields

I'm looking for an introductory text on valued fields, to be used as the basis for a reading group for model theorists. Currently, I know of one such text, Valued Fields by Engler/Prestel. However, ...
0
votes
0answers
32 views

Showing that $(\mathcal{M}, N) \equiv (\mathcal{N}, N)$

Let $T$ be a theory (complete and consistent) and let $\mathcal{M} \models T$. Let $\mathcal{N} \subseteq \mathcal{M}$. Suppose we have that $\exists \mathcal{M}' \supseteq \mathcal{M}$ s.t. ...
6
votes
3answers
90 views

Why can any type be realized?

I couldn't find this question asked previously, which means it's probably an especially daft question. Given an $\mathcal{L}$-structure $\mathcal{M}$, my textbook defines an $n$-type over $A\subseteq ...
2
votes
2answers
112 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
0
votes
2answers
128 views

Is there a useful application of Peano arithmetic?

If there is, can someone provide an example of how Peano arithmetic can be used to solve a real-world problem? If not, can someone provide an example of any axiomatic system other than ZFC that can ...
2
votes
1answer
44 views

Consistency first-order theories

is it true that any consistent first-order theory has a model? In case of affirmative response: 1) Is it the Godelian completeness proof? 2) Is there a standard strategy for constructing ...
4
votes
1answer
56 views

Ultraproducts by countably complete ultrafilter

I've recently learned about ultraproducts, but the source I learned from almost immediately after the definitions restricted to talking about countably incomplete ultrafilters. I know that the ...
4
votes
1answer
45 views

Which sentences survive the passage from $X$ to the set of all functions $I \rightarrow X$?

Suppose $X$ is a mathematical structure with a single underlying set which we will also denote $X$, equipped with some functions and relations. Letting $I$ denote an arbitrary non-empty set, we see ...
1
vote
0answers
59 views

Question about models theory

Which of the models M_1,M_2,M_3 in a picture is atomic? Which is saturated? For the two models that are not saturated, find 1-types which are omitted. For the two models that are not atomic, find ...
2
votes
1answer
53 views

Countably many worlds and Universal Sentence.

This is a naive, kind of informal argument. Suppose we have a language with just one predicate $P$ and constants $a_{1}, a_{2}, a_{3}$ and so on. Suppose also that we have countably many worlds $1, ...
2
votes
1answer
76 views

Is it true that $\Bbb Z^*\setminus\Bbb Z$ has no finite elements?

If we consider the hyperreals, we know that there exist non-zero infinitesimals so $\mathbb R^*\setminus\mathbb R$ has finite elements. However, it seems like that is not true for $\mathbb ...
2
votes
1answer
47 views

Comparing models through partial isomorphisms

Let $T$ be a theory of a language $\mathcal{L}$ with no function symbols. Let $\mathfrak{A}, \mathfrak{B} \models T$. For all finite sets $X \subseteq A$ and $Y \subseteq B$, there exists a function ...
2
votes
2answers
66 views

extension of group operation from $\mathbb{Q}$ to $\mathbb{R}$

I'm having a hard time with this (seems easy, but could be misleading) problem: Let $A \subseteq \mathbb{Q}$ be a convex subset, and let $+$ group operation on $A$. Let $\overline{A} := \{x \in ...
3
votes
2answers
81 views

Equivalence between two definitions of infinitary logic

The common definition of $ \omega $-logic (a.k.a $\mathcal{L}_{\omega_1,\omega}$ logic) is the usual first order logic allowing infinite conjunctions and infinite proof. Chang and Keisler, in section ...
0
votes
1answer
57 views

Models of the full theory of a structure

I'm reading Model theory: an introduction, by David Marker. I'm at page 14, where it says: ...one way to get a theory is to take $\operatorname{Th}(\mathcal{M})$, the full theory of an ...
6
votes
2answers
126 views

For two theories $T,T'$, what does $T\vdash Con(T')$ really tell us about the models of $T$?

Inspired by this question, which I realized I couldn't answer (because model theory and me don't get along). I've made a few edits to (hopefully constructively) tighten the question a bit. If for ...
6
votes
1answer
179 views

Using the Reflection Theorem

I've been reading about the Reflection Theorems in Kunen's 2011 Set Theory book. The idea that $ZFC \not \vdash \exists \gamma [V_\gamma \models ZFC]$, but $ZFC \vdash \exists \gamma [V_\gamma \models ...
0
votes
1answer
46 views

Omitting types theorem for types with parameters

Does the omitting types theorem as exposed e.g. in Hodges consider types with parameters or is it just about types over the empty set?
2
votes
1answer
42 views

Showing that $|\phi(\mathcal{N})| = \kappa$ s.t. $\mathcal{M} \equiv \mathcal{N}$ with $|\mathcal{N}| = \kappa$

Problem: Suppose $\mathcal{M}$ is an $L$-structure and $\phi \in L_n$ ($n > 0$) is such that $\phi(\mathcal{M})$ is infinite. Then show that for every cardinal $\kappa$ with $\kappa \ge |L|$ there ...
3
votes
1answer
124 views

Initial Segments of Modular Arithmetic

Modular arithmetic (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. MA is $\omega$-inconsistent and all infinite models ...
-4
votes
1answer
73 views

show $\equiv$ implies $\cong$ [closed]

Prove that for $L$-relational an arbitrary language, if $M$ is a finite $L$-structure and $M \equiv N$, then $M \cong N$. Do it for $L$ finite and then generalize it to infinite case. I am not able ...
0
votes
1answer
70 views

Showing that $\mathcal{M} \preccurlyeq \mathcal{N} \implies \mathcal{M} \equiv \mathcal{N}$.

Suppose that $\mathcal{M} \preccurlyeq \mathcal{N}$. Then by definition we have that $\mathcal{M}$ is a substructure of $\mathcal{N}$ s.t. for any (possibly empty) tuple $\overline{a}$ from $M^n$ and ...