2
votes
0answers
94 views

Isomorphism of finite models

Let $\mathfrak A$ and $\mathfrak B$ are models of finite signature $\sigma$. Prove that $\mathfrak A$ and $\mathfrak B$ are isomorphic, if $\mathfrak A \equiv \mathfrak B$ and $\mathfrak A$ is ...
1
vote
1answer
40 views

$(M, <) \equiv (\mathbb{Z}, <)$ and $(\mathbb{Q}, <)$ embeds into $(M, <)$

This is homework for a class I didn't take, but which is a prerequisite for a course I will take. In particular I supposed it's something other people may see as a homework problem in the future. ...
1
vote
1answer
37 views

Prove the elementary equivalence of the two models

There are two models $\mathfrak A$ and $\mathfrak B$ in class $K$. $\mathfrak A = <P(\omega), \subseteq>$ $\mathfrak B = <P(\omega), \supseteq>$ Is the $Th(K)$ of a full theory of ...
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 ...
1
vote
2answers
62 views

Logic: some basic plane geometry

Suppose you've got the language of some basic plane geometry, i.e. two 1-place relation symbols $P$ and $L$ for point and line and one 2-place relation symbol $I$ for point $x$ lies on line $y$. Now, ...
1
vote
0answers
99 views

the amalgamation property

Definitions: The age of a structure $M$ is the class of finitely generated substructures of $M$. A class of structures K has the amalgamation property (AP) if Whenever $A,B,C$ belong to $K$ ...
10
votes
0answers
221 views

Number of automorphisms of saturated models

I have the following assignment question: Let $M$ be an $L$-model of cardinality $\kappa$. Assume $M$ is saturated. How can you show that $|\text{Aut}(M)|=2^{|M|}$? I see two possible ...
3
votes
1answer
85 views

Classification of models

Let $L=\{P_0,P_1,P_2\}$ be a first order language, and let $$T=\bigg\{\Big(\forall x\ P_i(x)\Big)\vee \Big(\forall x\ \neg P_i(x)\Big):i \in \{0,1,2\}\bigg\}\\ ...
4
votes
1answer
72 views

Two homogenous structures realizing the same types are isomorphic

Let $M$ and $N$ be two countable, homogeneous structures, and assume that they both realize the same types with a finite number of variables. Does it follow that $M$ and $N$ are isomorphic? What if ...
1
vote
1answer
202 views

SHOW that there are infinitely many equivalence classes of formulas

Let $\mathcal{Q}$ denote the additive group of rational numbers, i.e. the structure $\left<\mathbb{Q}; +; 0\right>$. Let $\mathcal{L}$ be the language of $\mathcal{Q}$ and let $T$ be the ...
1
vote
2answers
69 views

Construct countable Boolean algebra

How can I construct a countably infinite Boolean algebra with $n$ atoms, for $n \in \mathbb{N}$?
1
vote
0answers
50 views

Axiomatizable classes

Are the statements below true or false: The class of finite sets is axiomatizable The class of infinite sets is axiomatizable The class of infinite sets is finitely-axiomatizable The class of fields ...
0
votes
1answer
71 views

Complete n-types of the theory of atomless Boolean algebras

I have to answer the next questions: What is the number of complete 1-types of the theory of atomless Boolean algebras? What is the number of complete 2-types of the theory of atomless Boolean ...
5
votes
1answer
100 views

The ultraproduct $\mathbb{N}^\mathbb{N} / \mathcal{F}$ is uncountable

I have to prove: Let $\mathcal{F}$ be a non-trivial ultrafilter on $\mathcal{P}(\mathbb{N})$. Prove that the ultraproduct $ \mathbb{N}^* = {\mathbb{N}^{\mathbb{N}}}/{\mathcal{F} } $ (I don't know if ...
6
votes
1answer
166 views

Non-isomorphic countable Boolean algebras

I'm trying to solve the next exercise: Construct a sequence $\mathcal{B}_0,\mathcal{B}_1, \ldots$ of countable Boolean algebras such that for all $m \neq n$ then $\mathcal{B}_0 \ncong \mathcal{B}_1$. ...
1
vote
0answers
37 views

Theory of (Q,+,0) has QE

I want to prove that $Th(\mathbb{Q},+,0)$ has quantifier elimination. Please point out where i go wrong or how to finish my reasoning: We can bring the formula in disjunctive normal form and ...
6
votes
1answer
70 views

Theory of (Z,+) has uncountably many 1-types

I'm working on some exercises in model theory, but on this one I don't know how to start. Please help to solve this. Prove that $\text{Th}(\mathbb{Z},+)$, the theory of the structure ...
2
votes
1answer
48 views

Expressing “uncountable” in $L_{\omega_1\omega}$

Given a countable signature $\tau$ I'm trying to find a uncountable $\tau$-Structure $\mathfrak{A}$ which does not satisfy the same infinitary logic $L_{\omega_1\omega}$-sentences as a countable ...
1
vote
1answer
55 views

All classes of finite structures are axiomatizable in $L_{\infty\omega}$

We want to proof that every class of finite structures is axiomatizable in the infinitary logic $L_{\infty\omega}$. We fix the signature $\tau$ (is okay to do so?). Thus, we can assume that for every ...
4
votes
1answer
80 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 ...
8
votes
2answers
118 views

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 ...
1
vote
1answer
133 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 ...
6
votes
1answer
138 views

The relation < is not definable in the structure of the integers with sucessorfunction

I have to prove the following statement: Prove that there is no formula $\psi=\psi(x_0,x_1)$ in the language $Th((\mathbb{Z},S))$ such that the relation $\left\{(m,n)\in\mathbb{Z}\times\mathbb{Z}: ...
9
votes
1answer
257 views

Compactness Theorem Application

I am doing an exercise on the compactness theorem of first order logic. The task is to prove that there is no singe first order sentence which is satisfied in exactly the infinite graphs (thereby, a ...
2
votes
1answer
56 views

Extend non-principal n-type

Is the following true? Let T be a complete theory in some elementary language. Let $n$ be a natural number and suppose $\Gamma$ is a non-principal $n$-type of T. Let $\Delta$ ne an $n+1$-type of T ...
1
vote
1answer
76 views

Prime Model and countable saturated model proof and types of a theory

I know that for the complete theory $Th((\mathbb{Q},+,0))$ we have the prime model $(\mathbb{Q},+,0)$ and the countable saturated model $(\mathbb{Q}^{\infty},+,0)$, but what should I do when I try to ...
3
votes
1answer
75 views

Prime Model and boundary functions for n-types

Find a prime model and a countable $\omega$-saturated model of $Th((\mathbb{Q},+,0))$. Define a function from $\mathbb{N}$ to $\mathbb{N}$ such that, for each n, $Th((\mathbb{Q},<))$ has not more ...
2
votes
3answers
146 views

Number of types in a given complete Theory

Prove that $Th((\mathbb{Q},<,+,0,1))$ has uncountably many 1-types. Prove that $Th((\mathbb{Q},+,0,1))$ has countably many 1-types. Prove that $Th((\mathbb{Q},<,0,1))$ has five 1-types. Prove ...
3
votes
2answers
97 views

First order sentence true in $\mathbb{Q}$ but not in $\mathbb{R}$.

I have the following assignment question: Find a first order sentence that is true in $\langle\mathbb{Q},+,\cdot ,0,1\rangle$ but not in $\langle \mathbb{R},+,\cdot,0,1\rangle$. Most of what I ...
2
votes
1answer
140 views

first order logic question model

suppose we have a model for a language in first order logic $ M=<D,I> $ such that D is the domain and I is the interpetation such that for every $ a \in D $ we have a closed noun (a noun with no ...
3
votes
2answers
157 views

dense linear orders DLO

I am asked to prove that if I have two models of dense linear orders DLOs WITH the minimum and maximum. must be izomorpic to each other by fining direct izomorphy. I seem to always get stuck ...
3
votes
1answer
225 views

finding n-types

My query is regarding following question:- Let $\mathcal Q$ denote the additive group of rational numbers, i.e. the structure $\langle Q ; +; 0\rangle$. Let $\mathcal L$ be the language of ...
1
vote
2answers
172 views

How to prove exact number of congruences over $\mathbb {R}$?

I have to prove that there are exactly 2 congruences over $\mathbb {R}$ seen as a model/structure $\tau = (\varnothing, {+,*}, \varnothing, \operatorname{arity}(+) = \operatorname{arity}(*)=2)$ where ...
6
votes
0answers
234 views

Using the compactness theorem to show a set of first-order formula is equivalent to a set of quantifier-free formula

I am going through some theorems in Hodges' ``A shorter model theory'' and I have realized that I do not understand a certain argument regarding compactness. My question has two forms, I am sure that ...
3
votes
2answers
97 views

$\Sigma_1 \cup \Sigma_2$ has a model

Let $\Sigma_1$ and $\Sigma_2$ be sets of $L$-sentences such that no symbol of $L$ occurs in both $\Sigma_1$ and $\Sigma_2$. Suppose $\Sigma_1$ and $\Sigma_2$ have infinite models. Then $\Sigma_1 \cup ...
6
votes
2answers
148 views

Axioms for atomless Boolean algebras

I'm embarrassed to be asking, but: "Write down a set of axioms for the theory of atomless Boolean algebras." This is Exercise 1.14 in Chapter 9 of "Models and Ultraproducts" by Bell and Slomson. I'm ...