# Tagged Questions

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### Every elementary submodel of $H(\aleph_1)$ is transitive

I am trying to solve the first part of Exercise II.17.30 in Kunen's Foundations of Mathematics which asks: Prove that every $A \preccurlyeq H(\aleph_1)$ is transitive. Here we are working over ...
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### There is no smallest infinitely large prime

I'm reading Kunen's Foundations of Mathematics and trying to solve Exercise II.16.19, which constructs an elementary extension of the reals as an ordered field, and asks the reader to prove various ...
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### Showing there does not exist a formal proof of a formula $\phi$.

My problem:Suppose $R$ is a binary predicate and use the soundness theorem to show that there does not exist a formal proof of $$\phi =\forall y\exists xR(x,y)\rightarrow \exists x\forall yR(x,y).$$ ...
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### 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 ...
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### $(M, <) \equiv (\mathbb{Z}, <)$ 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. ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 $K$ ...
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### 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 ...
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### 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\}\\ ...
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### 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 ...
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### 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 ...
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### Construct countable Boolean algebra

How can I construct a countably infinite Boolean algebra with $n$ atoms, for $n \in \mathbb{N}$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\mathcal{B}_0 \ncong \mathcal{B}_1$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...