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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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 ...
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535 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 ...
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115 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 ...
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78 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 ...
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329 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 ...
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226 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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332 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 ...
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234 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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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 ...
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129 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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204 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 ...
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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 ...
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130 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 ...
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220 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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159 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 ...
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62 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 ...
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535 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 ...
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1answer
283 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 $a_1, \ldots, a_n \in M$, ...
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113 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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106 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.
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151 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, ...
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284 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 ...
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138 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 ...
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219 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. ...
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285 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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100 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 ...
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224 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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206 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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268 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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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 ...
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1answer
146 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 ...
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2answers
342 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 ...
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359 views

Mathematical structures and signature

From Wikipedia: In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach (or relate) to the set, making it easier ...
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Question about the proof of consistency iff satisfiability of a theory

In pretty much any Model Theory or Logic textbook you will find the following claim, where $T$ is a theory (a set of $\mathsf{L}$-sentences), $T$ is consistent if and only if $T$ is satisfiable. ...
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1answer
247 views

Independence results in first-order PA and second-order PA

There are statements $\varphi$ that are independent of first-order Peano Axioms. Are these statements also independent of second-order Peano Axioms? I'm reading Wikipedia articles around ...
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0answers
52 views

Nice model theoretic properties of a theory adding a new predicate?

I would like to know what nice model theoretic properties (for example simplicity, NIP, stability, etc) can be preserved when we add a new predicate to the language. Explicitly, if $T$ is an L-theory ...
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1answer
400 views

Does the existence of a $\mathbb{Q}$-basis for $\mathbb{R}$ imply that choice holds up to $\frak c$?

The axiom of choice is, for ZF, equivalent to the statement that every vector space has a basis. The implication of AoC by the existence of a basis for any vector space is shown in this paper. The ...
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1answer
237 views

What does variety language mean?

In Zilber's paper entitled 'a theory of generic function with derivations', he works in the variety language for fields? What does this mean? If I want to define the variety language what can I say?
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182 views

I want examples of definable groups in algebraically closed fields?

I can not think of any example of a definable group in algebraically closed fields?
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272 views

Why is a commutative ring with an infinite number of idempotent elements unstable?

In a book of model theory I found the following statement: A commutative ring with an infinite number of idempotent elements unstable. I haven't manage to prove it yet. As stability in the model ...
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1answer
79 views

A question about $R^\infty$-rank

I have been trying to learn some stability theory lately, and I have been reading "Geometric Stability Theory" by Pillay. There is a lemma he states without proving which I am trying to prove, but I ...
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148 views

Superstructure with sets as atomic/base entities

This question arose while learning nonstandard analysis. The superstructure $V(X)$ of a nonempty set $X$ is defined recursively: $$\begin{align*}V_0(X) &= X \\ V_{i+1}(X) &= V_i(X) \cup ...
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1answer
62 views

Modelchecking on Automata, $\phi$ not SAT and $\phi \models$ False

Given a formula $\phi$ Is $\phi \models FALSE$ equivalent to $\phi$ not SAT? Or does $\phi \models FALSE$ means that $\phi$ is never $TRUE$ and $\phi$ not SAT means, that there existst at least one ...
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270 views

Consistency of PA: why other proofs?

Completeness theorem affirms that a formal first order system is consistent iff it has a model. The FOL number theory(PA) or First Order Arithmetic has a model, which is the natural numbers structure. ...
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335 views

Why does creating a model show consistency?

As per the title, why does the ability to generate a model from axioms prove they are consistent?
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207 views

Characterization of totally categorical theories

I have what I am sure is a trivial question, but I can't seem to answer it for myself. In model theory, there is a theorem of Hrushovski which shows that if T is a totally categorical theory (i.e., T ...
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2answers
190 views

Conservative extension is equiconsistent?

Is it true that a conservative extension of a theory is equiconsistent with it, and, if so, why? WP says: "In mathematical logic, a logical theory T2 is a (proof theoretic) conservative extension of ...
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390 views

How exactly do logic and mathematics interact and what happens when we change the logic?

[Note: this question turned out to be pretty huge, so if you think it would be better to split it up into smaller questions, please comment. The questions here are quite conceptually intertwined and ...
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234 views

Can locally finite structures really be defined as those structures omitting a certain quantifier-free type?

My question is 2.3.8(b) in Hodges' A Shorter Model Theory, p.43. (It's also in his original Model Theory on p. 47). I'll cite the problem, follow it by definitions of key terms, and then explain why ...
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173 views

number of linear orders

It is well known that for every infinite cardinal $\kappa$ the number of non-isomorphic total orders of cardinality $\kappa$ is $2^\kappa$. Who first proved this, and in what context? Was it proved ...