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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the solution of ode can be encoded as first order logic

Consider the following sytem of ODEs $\dot{x}= Ax$, and given $x(0)$, where $A$ is a $n\times n$ matrix with rational entries. Can I encode the solution, say $x(t)$ for a given $t$, as a first ...
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2answers
138 views

Am I allowed to realize one object twice within one set-theory?

Say I consider a set theory with the Axioms of Extensionality and the Axiom of Pairing. As I understand it, stating the axiom allows me to make a definition like $$(a,b):=\{\{a\},\{a,b\}\}$$ and ...
3
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2answers
217 views

Why is completeness theorem true?

Who can teach me completeness theorem? Thanks! Recommending a book is also welcome. More specifically, it says that if a statement is true in all models of a theory, then it has a proof from this ...
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145 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 ...
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1answer
107 views

Absoluteness and categories

From the wikipedia article on the Skolem paradox: A central goal of early research into set theory was to find a first order axiomatisation for set theory which was categorical, meaning that the ...
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142 views

Saturated Boolean algebras in terms of model theory and in terms of partitions

Let $\kappa$ be an infinite cardinal. A Boolean algebra $\mathbb{B}$ is said to be $\kappa$-saturated if there is no partition (i.e., collection of elements of $\mathbb{B}$ whose pairwise meet is $0$ ...
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2answers
83 views

Logic question proving something about compactness

Let $\Sigma$ be a set of formulas. There's a finite set $\Lambda \subseteq \Sigma$. I'm asked to prove or disprove that $\Sigma$ has a model if and only if $\Lambda$ has a model. It seems to me ...
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9answers
714 views

Learning Model Theory

What books/notes should one read to learn model theory? As I do not have much background in logic it would be ideal if such a reference does not assume much background in logic. Also, as I am ...
2
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2answers
72 views

Embedding of standard model of arithmetic to PA-model

I am working on the following problem: Let $ S_{Arithmetic} = \{+, *, 0, 1\}, \mathfrak{M} $ a model for PA (first-order peano axioms) }, and $ \mathbb{N} = (\mathbb{N},+ ^{\mathbb{N}}, ...
4
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1answer
208 views

Lindenbaum algebra is a free algebra

The following is a continuation of this question. I would like to prove that the Lindenbaum algebra is a free algebra. Hopefully I would like to hear hints on how to proceed in the 'right' ...
4
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1answer
148 views

Are there simple counterexamples to a strengthening of omitting types theorem

The famous Ehrenfeucht's omitting types theorem states that for any countable set of nonisolated types (without parameters), there is a (countable) model such that it does not realize any of them. A ...
3
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1answer
297 views

Complete/incomplete theory

I am thinking about completeness and incompleteness of theory's, and to illustrate both properties i am thinking of how to build an complete system, and then turn it into an incomplete one. Example. ...
2
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1answer
180 views

Is there a relationship between the Compactness Theorem and the upward Lowenheim-Skolem Theorem in FOL?

Is there a relationship between the Compactness Theorem and the upward Lowenheim-Skolem Theorem in FOL? I was thinking of another post of mine "Why accept the axiom of infinity?" when I though, ...
2
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0answers
137 views

Can we find a nice definition of Congruence in Topology?

According to my knowledge, quotient structure is a original structure divided by a congruence. However, quotient topology space is defined this way. Quotient_topology In this way, $\sim$ is only said ...
0
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1answer
82 views

Relations of language/theory/signature

Say that the language of the first order logic is the collection of symbols that can be used in the formulas + the grammar (the rules that specify how they can be combined)? 1) However, the signature ...
3
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1answer
160 views

Explanation of how models can differ on $\omega$?

Assuming set theory (here, ZF) is consistent, there is a model $V$ of ZF, the universe of all sets. So, there is a $\omega^V\in V$. A set $A\in V$ is countable iff a bijection $f\in V$ exists ...
3
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2answers
167 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 ...
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2answers
123 views

What's the name of this operator?

Let $f,g$ be functions in $C^A$ and $C^B$ respectively. Let $\boxtimes:C^A \times C^B \to (C\times C)^{A \times B}$ s.t. $f\boxtimes g(a,b)=(f(a),g(b))$ It seems not the tensor product, nor ...
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1answer
83 views

How to formally describe this Uppaal automata?

I have the following simple automata: What I'm looking for is a formal description of this based on the definition here $A=(\Sigma,\Gamma,S,s_0,\delta,\omega, F)$ How to declare all the ...
0
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1answer
325 views

Models of propositional logic

Define a theory of propositional calculus as the set $T$ of axioms (expressed in propositional calculus) and a set of valid symbols. What I would like to see are some examples of theories in ...
2
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2answers
108 views

Does Bernstein theorem hold for models with elementary functions?

Bernstein theorem is a general pattern that occurs in many areas of mathematics (see the Wikipedia article for some examples). Does it hold for arbitrary models with elementary embeddings? To be more ...
0
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1answer
71 views

Factor in ultraproduct

The general method for getting ultraproducts uses an index set I, a structure $M_i$ for each element i of I (all of the same signature), and an ultrafilter U on I. The usual choice is for I to be ...
2
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2answers
270 views

confusion regarding compactness theorem

I am getting somehow confused of compactness theorem. The compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This ...
0
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3answers
203 views

Proof that theory of infinite numbers does not have finite model

Just curious: Is there any proof that proves that there is no finite model of infinite numbers (Theory of infinite numbers)? Edit: by infinite number, I mean either Reals or Naturals.
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4answers
274 views

Model of theory of real closed field

I heard somewhere that models of theory of real closed field are isomorphic. However, there is also a statement in Internet which seems to say the opposite. Are the models of theory of reals ...
2
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1answer
223 views

Countability in first-order logic is relative to what exactly?

Skolem's Paradox tells us that countability in first-order logic is relative. Relative to what? Below is what I've gathered. Countability it relative to: 1. what a model takes to be $\mathbb N$ 2. ...
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4answers
1k views

Tarski's decidability proof on real closed field and Peano arithmetic

It seems very confusing that real closed field (which also can be used as the theory of real number) is decidable, while Peano arithmetic, which seems to be a subset of real closed field is ...
3
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1answer
117 views

Interpretation of nonlogical symbols in compactness arguments

Please don't give me a complete answer to the motivation part of the question. I want to figure that part out for myself. Motivation: As a starting example, say that a reversing function is a unary ...
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3answers
178 views

A quick question about categoricity in model theory

I just want to see if I am using the term "categoricity" correctly in the following context: (1) I was thinking about why someone might reject a simple resolution to Skolem's Paradox. (2) The ...
0
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1answer
94 views

Distinction between the universe of a model and the domain of a model?

Distinction between the universe of a model and the domain of a model? I'm pretty sure I'm wrong about this. But even reading Wiki, I'm still not clear. I'll use an example to illustrate what I ...
4
votes
1answer
121 views

expressiveness of computable infinitary logic

A language $L_{\omega_1\omega}$ generalizes an ordinary first-order language by allowing countably long disjunctions. If we take its nonlogical vocabulary to contain just a predicate for the ...
4
votes
3answers
215 views

coproducts of structures

Suppose $S$ is a family of $L$-structures where $L$ is some collection of constant symbols, relation symbols, and function symbols. Does the coproduct of elements of $S$ exist? If not, how does one ...
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2answers
128 views

What are the possible minimal acl-dimensions of strongly minimal models?

The question is as in title. By acl-dimension I understand the cardinality of maximal acl-independent set (well-defined for strongly minimal theories). By minimal I understand that there is no ...
5
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1answer
60 views

Mutual Uniqueness of Operations in PA models

Let us consider the first-order theory of Peano arithmetic (from now on PA) formulated in the vocabulary with just $+$ (for addition) and $\cdot$ (for multiplication). This vocabulary restriction is ...
4
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1answer
410 views

advantage of first-order logic over second-order logic

As I look over the post that has the similar question, I began to wonder: The only reason I found is that first-order logic can prove validity of some second-order logic formula/sentences, as some of ...
2
votes
1answer
242 views

Proof of the Löwenheim-Skolem theorem

For each first-order $\sigma \,$-formula $\varphi(y,x_1, \ldots, x_n) \,,$ the axiom of choice implies the existence of a function $f_\varphi: M^n\to M$ such that, for all $a_1, \ldots, a_n \in M$, ...
4
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3answers
106 views

Can the truth value of an independent property be changed at will by enlarging the model?

Let $\phi$ be a property that's independent of $ZFC$, so that there are strcutures ${\mathfrak A}=(A,{\in}_A)$ (where $A$ is a set or class and ${\in}_A$ is a binary relation on $A$) that are models ...
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2answers
103 views

Difference between axiomatization and model

As I study through set theory, I find the definition of axiomatization and models somewhat confusing. The question is what is the difference between axiomatization and model? Thanks.
1
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1answer
133 views

Lowenheim-Skolem theorem confusion

In Wikipedia ( http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem ), it says: In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, ...
3
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1answer
234 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
1answer
128 views

Can the ongoing need for a meta language be stopped by a loop?

As an afterthought to this question on sets in set theory, and more specifically to the observation that a (first-order) logic requires a meta-language to explain itself (i.e. there is already an ...
8
votes
1answer
204 views

$\aleph_1$-categorical fields are algebraically closed.

I'd like to understand the proof that if $K$ is an infinite field the theory of $K$ is $\aleph_1$-categorical, then $K$ is algebraically closed--but I'm having trouble finding it in the literature. ...
4
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1answer
255 views

how to show that a group is elementarily equivalent to the additive group of integers

Is there any fairly easy way of showing a group is elementarily equivalent to the additive group of the integers? I've found a simple characterization here: A ‘natural’ theory without a prime model, ...
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1answer
92 views

Proof through consistency

Take first-order Peano Arithmetic PA. We know that Gentzen proved PA consistent. Now, if one sets for example $\varphi$ to represent Fermat's theorem in FO, would proving PA+$\varphi$ consistent be ...
3
votes
3answers
181 views

Non-Archimedean non-standard models for R

Let $\langle R,0,1,+,\cdot,<\rangle$ be the standard model for R, and let S be a countable model of R (satisfying all true first-order statements in R). Is it true that the set 1,1+1,1+1+1,… is ...
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2answers
204 views

Is this incompleteness result easier to get than incompleteness of PA?

Gödel's theorem for Peano Arithmetic shows that (under consistency hypothesis on PA) there is a statement which cannot be proved or disproved within PA that is true under the standard model ...
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230 views

Theories and models

I apologize if my question is not well formed. The reason for it is that I don't understand the concepts enough to be able to ask a completely meaningful question. In the classes we said that a ...
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3answers
2k views

Is Foundational Research a Dead Field?

I'm a second year mathematics major at a pretty good school. Ever since I became a math major I have been most interested in set theory and logic, which I guess can be lumped into the category of ...
3
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1answer
120 views

Automorphisms of elementary extensions

I think this is probably a very simple question, but I've been puzzling over it for a while and can't seem to get anywhere. Suppose $M$ is a structure, $\alpha$ is an automorphism of $M$, and $N$ is ...
2
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2answers
295 views

Lowenheim-Skolem theorem and first-order model

In Wikipedia, it says that a nonstandard model of natural numbers is not first-order. But, from the Lowenheim-Skolem theorem, I don't see anything that points to this conclusion. Can anyone show me ...