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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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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1answer
51 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 ...
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1answer
47 views
Uncountable models for a language $L_Q$
$L$ is first-order language with identity and $L_Q$ a language obtained by adding to $L$ the quantifier $Q$.
Definition of $Q$: If $P$ is a formula and $x$ a variable, $QxP$ is a formula of ...
7
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1answer
62 views
What techniques are available for “surgical adjustment” of models of set theory?
Suppose I have a model $M$ of set theory (ZFC, or whatever). Let's say that I want to take a set $a$ out of it, and still have a model of set theory. For the sake of argument, say $a$ is one of the ...
5
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1answer
86 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 ...
5
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0answers
76 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 $\mathcal{B}_0 \ncong \mathcal{B}_1$.
...
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2answers
69 views
The standard approach to second-order axiom systems
This is a very basic question, but for some reason I couldn't find an answer elsewhere on the Internet.
Suppose we have an axiom system $A$ written in the language of second-order logic. In order to ...
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34 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
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1answer
53 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 ...
3
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1answer
31 views
Equivalent Definitions of Types
In my reading, I have seen two different definitions of an $n$-types which I think are equivalent, but I am stuck in showing this.
First I will fix notation:
Let $\mathcal{M}$ be an ...
3
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1answer
76 views
Formally Developing Model Theory Within ZFC
After reading Kunen's section on Model Theory in his Set Theory (2011), I'm still unsure about how model theory is formally developed within ZFC.
I was under the impression that all the symbols ...
3
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1answer
79 views
Tarski-Vaught test for $\preceq$ reduction
The Tarski-Vaught test for $\preceq$ states that given a structure $\mathfrak{B}$ and $A\subseteq B$ then $A$ is the underlying set of an elementary substructure of $\mathfrak{B}$ iff for all formulas ...
3
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1answer
149 views
Compactness for definable models?
In this question, I use "model" to mean a model in the language $\{\in\}$ of set theory. Call a model $M$ "definable" iff for every $x \in M$, there is a formula $\phi(\vec{y},z)$, where $\vec{y} \in ...
4
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1answer
120 views
Two forms of Beth's theorem?
The version of Beth's theorem I'm familiar with is that if $\phi$ is a sentence in the language $\Sigma\sqcup \lbrace R\rbrace$ depends only on $\Sigma$ (i.e., ...
6
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2answers
125 views
Is the completeness theorem for first-order logic relative to one's choice of set theory?
By the completeness theorem for first-order logic, every consistent theory has a model. However, to even make sense of the word "model," I believe we're assuming a set theory. So is there a set theory ...
6
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2answers
109 views
Show that an elementary equivalent structure of $\mathbb{R}$ and definable from points is Dedekind complete
On page 103, A Mathematical Introduction to Logic, Herbert B. Enderton（2ed),
Assume that $\mathfrak {A} \equiv \mathfrak R$ ($\mathfrak {R} = (\mathbb{R},<,+,\cdot)$). Show that any subset of ...
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0answers
83 views
Models of infinite cardinal and compactness
I'm stuck with this problem:
$L$ is first-order language with identity and $L_q$ a language obtained by adding to $L$ the quantifier $Q$. If $P$ is a formula and $x$ a variable, $QxP$ is a formula ...
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1answer
72 views
Prove that the following logical implication is true
Taking advantage of the model theory, prove that
$M \cong N \implies M \equiv N$
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1answer
82 views
m-categoricity of theory of algebraically closed fields of fixed characteristic
one more question on a point on page 89. At first glance, I accepted the result with respect ACF(n) shown in (iv). But now I wonder: Is Shoenfield referring to a result of model theory? I thought ...
1
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1answer
41 views
How often do incomplete types meet the hypotheses of the omitting types theorem?
I find the following formulation of the hypothesis (namely, non-isolation) for the omitting types theorem. A type $p$ over $T$ is "isolated" iff there is a formula $\phi(\vec{x})$ such that $\exists ...
3
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2answers
102 views
Model theory in group theory
I am interested in useful results for group theorists that can be shown using model theory. For example :
Theorem: Let $\langle X \mid R \rangle$ be presentation of a group $G$ with $X$ finite and ...
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1answer
183 views
why algebraic structures?
According to wikipedia, an algebraic structure is an arbitrary set with one or more finitary operations defined on it. From a model theory perspective, I understand this definition as: structure with ...
7
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1answer
118 views
Non-Isomorphic Ultrapowers
It is clear that given a family $(\mathfrak{A}_i)_{i\in I}$ of $L$-structures their ultraproduct may depend on the choice of the ultrafilter (for this question I am only considering non-principal ...
3
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1answer
68 views
Quantifier Elimintion of $(\Bbb{Q},+,0)$
I want to prove that the structure $(\Bbb{Q},+,0)$ has Quantifier Elimination.
I can prove it for some simple basic formulas, but what if i get a formula which says that i have a linear combination, ...
2
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1answer
33 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 ...
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1answer
45 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 ...
3
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2answers
58 views
Class of structures isomorphic to $(\mathbb{Z},<)$ in infinitary logic $L_{\omega_1 \omega}$
We want to define the class of all structures isomorphic to $(\mathbb{Z},<)$ in the infinitary logic $L_{\omega_1 \omega}$. Therefor we define strict order as usual:
$\forall x,y,z ~~ (x<y ...
2
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2answers
83 views
Are all models of peano arithmetics descibed using first order logic non standard?
It is known that there are non-standard models of Peano Arithmetics when it is described using first order logic. My question is if there is standard model (one which does not contains non-standard ...
2
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1answer
151 views
Paradox: Any set theory without universe set is not a model of itself
Because a model of a first order theory is not allowed to use a proper class as its domain, we can't use the universe of the set theory from the "meta-level" directly as a model for a first order ...
2
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1answer
88 views
Model theory for intuitionistic predicate logic: non-empty domain?
In classical logic we tend to make the assumption the domain of quantification is non-empty. This isn't (too) problematic because classical mathematicians assume a language/mind/proof independent ...
2
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1answer
115 views
How to prove that something is definable or not definable in a given structure?!
Hello friends of mathematics :)
I have some problems with the topic "Is something definable in a structure". I can solve some problems for example the following questions:
Is the relation definable ...
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1answer
53 views
Quickie on Boolean valued models
Bell writes on page 21 (you may use the search in the preview to search for "21" to view the page):
"..., we show that, for any complete Boolean algebra $B$, all the theorems of $ZFC$ are true in ...
3
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2answers
106 views
Question about passage in Halbeisen's book
I am looking at the following passage in Halbeisen's book "Combinatorial Set Theory" (p 260 at the bottom):
What is the role of $\Phi$? It seems to me that a finite fragment is the same as a ...
2
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1answer
91 views
Question about the proof of $GCH$ holds in $\mathbf L$
I have a question about the proof of the following:
(Lemma 37) Assume $\mathbf V = \mathbf L$, and let $\kappa$ be a cardinal. Then
$\mathcal P (\kappa ) \subseteq L_{\kappa^+}$.
Assume we ...
12
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2answers
162 views
Applications of model theory to analysis
Some of the more organic theories considered in model theory (other than set theory, which, from what I've seen, seems to be quite distinct from "mainstream" model theory) are those which arise from ...
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2answers
167 views
Example of a set that is in $\mathbf V$ but not in $\mathbf L$
Let $\mathbf V$ denote the cumulative hierarchy and let $\mathbf L$ denote Gödel's constructible universe. We then have $\mathbf L \subseteq \mathbf V$.
Would someone give me an example of a set that ...
2
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1answer
52 views
Possible typo in Just/Weese's set theory
In Just Weese on page 197 there are the following corollaries:
Regarding Corollary 24: Is this a typo and should say "$CON(ZF) \not\rightarrow CON(ZF + \exists \text{ "a strongly ...
2
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1answer
71 views
Understanding the model-theoretic proof of Hilbert's Nullstellensatz
The proof I am talking about goes like this: Given $k$ algebraically closed and $(f_1,..,f_k)=I\neq (1)$ an ideal in $A=k[x_1,..,x_n]$, let $m$ be a maximal ideal with $I\subseteq m$ and observe that ...
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1answer
72 views
Proving Axiom Schema of Replacement holds in $H_\lambda$
I think I proved the following, can you tell me if my proof is correct?
Exercise 23: Show that if $\lambda$ is a regular infinite cardinal then $\langle H_\lambda , \overline{\in} \rangle$ satisfies ...
2
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3answers
101 views
How to break power set for non-transitive models?
Apparently, $\varphi (x,y) : y = P(x)$ where $P$ denotes the power set is not absolute for transitive models.
We call a formula $\varphi(v_1, \dots v_n)$ in $L_S$ absolute for a class $\mathbf X$ ...
6
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3answers
124 views
Model Theory and Topology Connections
I have studied a bit of model theory, when I say "a bit" I have studied much more than is available to a typical undergraduate in the UK (i think, certainly from what I have seen) but I am sure this ...
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3answers
77 views
Extensionality of a relation and Axiom of Extensionality
What's the difference between the Axiom of Extensionality $(A1)$ and an extensional relation?
The definitions are
$(A1) \forall x,y ( x = y \leftrightarrow \forall z ( z \in x \leftrightarrow z \in ...
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4answers
136 views
A question about standard models
As I understand it a standard model is a model in which the relation is the $\in$ on the actual set of sets constituting the model.
(i) Hence theories that aren't in the language of set $L_S$ ...
3
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0answers
124 views
A question about a passage in Just/Weese's Basic Set Theory
I have a question regarding the following passage in Just/Weese (p 192), $\mathbf{V}$ denoting the cumulative hierarchy:
"But consider the following situation:
Where in "($\beta$)" does the ...
3
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1answer
71 views
Question about cumulative hierarchy
In the following let $\mathbf{V} = \bigcup_{\alpha \in \mathbf{ON}} V_\alpha$ denote the cumulative hierarchy. Let $\{\varphi_0, \dots, \varphi_n, \dots \}$ denote a list of all $ZF$ axioms. I am ...
2
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1answer
149 views
Can we prove the completeness of FOL based on forcing?
In David Marker's Model Theory, there is a exercise said " we can view a countable Henkin construction as a forcing construction ". But in the book the Henkin construction is used to prove the ...
2
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0answers
36 views
Model complete theories of henselian local rings which are not nec valuation rings
I just want to ask if anybody as any examples of a first order model complete theorie of henselian local rings which is not some theory of valuation rings. More precisely-
I am looking for a theory ...
2
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1answer
90 views
Does quantifier elimination imply decidability?
I know in some simple cases of quantifier elimination that I have seen, one ends up seeing that the process of quantifier elimination resulted in being able to show decidability. Is this true in ...
3
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3answers
109 views
Is every model of PA well-ordered?
Assume first-order Peano arithmetic is consistent and $N$ is its model, we know that every subset of $N$ contains a minimal element. It's a second-order property so I am not sure if it hold in ...
11
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3answers
175 views
Does every complete theory admit quantifier elimination?
Does every complete theory admit quantifier elimination? I know that at least in some simple cases the reverse is true; such as some reducts of number theory.Thanks
