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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247 views

Ehrenfeucht–Fraïssé game, how can I understand it?

My course of "Formal Methods" deals with Ehrenfeucht–Fraïssé games, particularly regarding the inexpressibility in FO logic. At the moment I've fully understand what this games are and how they are ...
0
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1answer
104 views

Question on the ultrapower construction of superstructure

On page 83-85 (here, on googlebooks), An Introduction to Nonstandard Real Analysis, Albert E. Hurd, Peter A. Loeb, two steps are given to construct a monomorphism $\ast : V(X) \to V({}^{\ast} X)$. The ...
0
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1answer
40 views

Signatures and L-Structures

Consider the field of real numbers $\mathbb{R}$. This is an $L$-structure. Is there such a thing as an $S$-structure (i.e. a signature structure)? Or because we can recover a first order language from ...
7
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3answers
206 views

Is there any countable ordinal number which has a member undefinable?

Let $Q\colon=\{\alpha \in Ord \mid \exists \beta \in \alpha(\beta \text{ is undefinable in } \langle\alpha,<\rangle )\}$ denote the set of all ordinal numbers which has an undefinable member. ...
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3answers
72 views

Show $s(s(a))=s(b)$ implies $s(a)=b$

Let us have a first order language $L=\{0,s\}$, where $0$ is a constant, $s$ is a function symbol of arity $1$. The first-order theory $T$ is axiomatized as follows: $\forall x \neg( s(x) = 0)$ ...
4
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1answer
267 views

A sentence that has infinite models, finite model, but no finite model above certain cardinality

Let $T$ be a theory and $\sigma$ a sentence, such that there exists infinite $\mathfrak{A} \models T + \sigma$. there exists finite $\mathfrak{A} \models T + \sigma$. there exists $n \in ...
3
votes
1answer
98 views

What's the idea behind the proof of saturation of internal sets via ultrapower construction?

I'm trying to understand the proof of saturation of internal sets via ultrapower construction on Robert Goldblatt's Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. Though, it's ...
2
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1answer
66 views

Can I express (only) the syntactical formulation of a set theory within another.

Motivated by this question on the SE philosophy board "Is there a one true set theory", I want to ask for a clarification: I've seen e.g. ZF being expressed in Grothendieck set theory. If I say ...
1
vote
1answer
72 views

Every connected $\omega$-stable group has a zero element?

Let $G$ be a connected $\omega$-stable group and $p$ its unique generic. Let $a$ be a realization of $p$, $G\prec G_1$ an elementary extension containing $a$ and $q$ the non forking extension of $p$ ...
5
votes
1answer
77 views

Point evaluation of a linear functional on an Ultrapower

Let $E$ be a Banach space and $(E)_{\mathcal U}$ be an ultrapower for some ultrafilter $\mathcal U$ on an index set $I$. It is remarked in a paper that $(E')_{\mathcal U}$ can be naturally embedded ...
5
votes
2answers
374 views

Do isomorphic structures always satisfy the same second-order sentences?

I know that if two mathematical structures are isomorphic, then they satisfy the same first-order sentences. The converse is false. This is probably a completely obvious question, but is it true that ...
10
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1answer
255 views

Is this a characterization of well-orders?

While grading some papers and thinking about a question related to well-orders (in particular, pointing a mistake in a solution), I came to think of a reasonable characterization for well-orders. I ...
6
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1answer
127 views

Model Theory (Hodges), Section 2.1, Exercise 13

Let $K$ be a field of characteristic 0, $n$ a positive integer, and $G$ the group $GL_n(K)$ of invertible linear transformations on $K^n$. Show that the following subsets of $G$ are ...
2
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1answer
122 views

Not Skolem's Paradox

Assume we have a countable, non-standard model of Peano Arithmetic (PA) in ZFC. http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic Let $N^*$ be the universe of this model and let $m \in ...
3
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0answers
155 views

Crankery: Is there a perfect inner model of ZFC?

In the book "Aspects of Vagueness", the article "The alternative set theory and its approach to Cantor's set theory" by A. Sochor proposes the following definition: We will say that a set universe ...
5
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1answer
55 views

Restrictions of automorphisms to elementary substructures

Suppose that I have structures $M \preceq M'$ (in some first-order language). I have a set $A$, with $M \subseteq A \subseteq M'$, and an automorphism $f$ of $M'$. Is it is always possible to find ...
144
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1answer
4k views

Is there a 0-1 law for the theory of groups?

For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } ...
4
votes
1answer
124 views

Definiteness of omega

A homework(ish) problem from models of set theory: Define $\varphi(x) :\leftrightarrow Lim(x) \land \forall y\in x \, (Lim(y)\rightarrow y=0)$ where $Lim(x)$ means that $x$ is a limit ordinal. ...
0
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1answer
149 views

non-axiomatizable logics

Hope you're all doing well. My question is about non-axiomatizable logics. My understanding is that a "logic" (the mathematical structure) is just another word for a "propositional calculus" as in ...
5
votes
1answer
65 views

Omitting Types… recursively

I'm working on the following problem at the moment: Let $\mathcal{L} = \{R\}$, where $R$ is a binary relation symbol. Let $T$ be a consistent, decidable $\cal{L}$-theory, and let $p(x)$ be a ...
3
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1answer
64 views

Complexity of index sets in nonprincipal ultrafilters

Let $U$ be a nonprincipal ultrafilter on $\omega$. It can be shown that the set $I = \{e \mid W_e \in U\}$ (where $W_e$ is the $e$th r.e. set in some given enumeration) cannot be $\Delta_2^0$ (in ...
1
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1answer
135 views

Standard models being non-standard?

If there is a ''set'' W in V which is a standard model of ZF, and the ordinal κ is the set of ordinals which occur in W, then Lκ is the L of W. If there is a set which is a standard model of ...
3
votes
0answers
98 views

generating complete consistent theories

This is a model theoretic question. I was reading Kremer & Mints's Dynamic topological logic paper, and it mentioned that by a “standard argument”, every consistent formula is a member of some ...
3
votes
2answers
106 views

First order logic, finite partially ordered structure problem

The exercise below is part of an exam for a Logic course. There were other questions leading up to this, but I've included their "result" in the wording below. Suppose alphabet $\mathcal{L} = ...
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0answers
115 views

Is this first order version of the Collatz conjecture decidable in peano arithmetic?

Let $\phi(x)$ be a first order formula in the language of arithmetic with one free variable $x$. Consider the sentence $\psi_\phi$, defined as: $$\phi(0)\wedge \phi(1) \wedge (\forall x \phi(x) \to ...
4
votes
1answer
204 views

Proving Tychonoff's theorem with the Compactness theorem of logic

It seems to be known that Tychonoff's Theorem for Hausdorff spaces and the Compactness theorem of first order logic are both equivalent over ZF to the ultrafilter lemma. Does anyone know a slick proof ...
3
votes
1answer
203 views

What's correspondence between the model theoric and the set theoric kernel of homomorphism?

A kernel of a mapping $h$ from $\mathfrak{A}$ to $\mathfrak{B}$, generally, is an equivalence relation $\{(a,a') \in \mathfrak{A} \times \mathfrak{A} \mid h(a)=h(a')\}$. However, in model theory, ...
3
votes
1answer
89 views

Automorphisms between two tuples of the same type

Fix some first order structure in some fixed language. If tuple $b$ is the image of tuple $a$ under some automorphism, then they have the same type. Is the converse true? That is, for any fixed tuples ...
3
votes
1answer
62 views

Can saturation drop when passing to an elementary extension.

Suppose I have a model $\mathcal{U}$ with some theory $T$ such that $\mathcal{U}$ is $\kappa$-saturated for some infinite cardinal $\kappa$. If $\mathcal{V} \succeq \mathcal{U}$ is an elementary ...
15
votes
1answer
365 views

A *finite* first order theory whose finite models are exactly the $\Bbb F_p$?

Since this question turned out to be trivial, I'm now asking this strengthened version: Is there a finitely axiomatized first order theory $T$ in the language of rings such that its finite models ...
3
votes
1answer
158 views

Question on the non-existence of a satisfaction formula in $\mathbb{L}$

I know this may sound trivial, but I'm having trouble figuring out what the problem is. Let $\mathbb{L}$ be the class of constructible sets. We know that $(\mathbb{L}_{\omega + \omega}, \in)$ is a ...
7
votes
1answer
147 views

A first order theory whose finite models are exactly the $\Bbb F_p$

Is there a first order theory $T$ in the language of rings such that its finite models are exactly the fields $\Bbb F_p$ with $p$ prime (but no $\Bbb F_q$ with $q$ a proper prime power is a model of ...
0
votes
1answer
209 views

first-order logic: Any finite set of first-order sentences true in all divisible abelian groups is true in some non-divisable one.

The question is the Subject. This comes from Barwise: Intro to First Order Logic, it is prop 2.1 and it is answered 2 pages later. I however don't understand the proof well. As Barwise says, the ...
5
votes
1answer
183 views

Why is there apparently no general notion of structure-homomorphism?

In model theory, one typically defines only embeddings of structures and isomorphisms, but I haven't seen a definition of general structure homomorphisms. Is there some particular reason behind that? ...
10
votes
1answer
139 views

Is there a universal property for the ultraproduct?

Given an ultrafilter U on a set I and a collection of ser X_i ($I \in I$) one defines the ultraproduct as the quotient of $\prod X_i $ by the identification $x_i=y_i :\leftrightarrow \{i:x_i=y_i\} \in ...
8
votes
5answers
1k views

What is exactly the meaning of being isomorphic?

I know that the concept of being isomorphic depends on the category we are working in. So specifically when we are building a theory, like when we define the natural numbers, or the real numbers, or ...
3
votes
1answer
105 views

Forking Independence

Marker proves in "Model Theory" that for an $\omega$-stable group, $G$, if it is connected then it has a unique generic type. I'm trying to understand the proof. The proof goes like this: Let $p,q$ ...
11
votes
2answers
567 views

Is Gödel's incompleteness theorem provable without any model-theoretic notion?

The entry on Gödel's incompletenss theorem in Wikipedia says: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, ...
4
votes
1answer
93 views

Are there other, non-model theoretic, accounts of semantic validity?

When I first encountered the double turnstile ($\models$) in the context of an intro to logic class it was presented to me as expressing semantic validity where this was cashed out model-theoretically ...
6
votes
2answers
222 views

Henkin vs. “Full” Semantics for Second-order Logic and Multi-Sorted First Order Interpretations

In this paper by Jeff Ketland, he notes: With Henkin semantics, the Completeness, Compactness and Löwenheim-Skolem Theorems all hold, because Henkin structures can be re-interpreted as many-sorted ...
4
votes
1answer
92 views

Class of finite groups a Fraïssé Class? [duplicate]

Is the class of finite groups a Fraïssé class? Calling this class $K$, does $K$ satisfy the following: Joint embedding property Amalgamation property Hereditary property: if $G \in K$ and $H \le G$, ...
4
votes
1answer
55 views

Field extensions of $\prod \Bbb F_p /U$

The ultraproduct of all finite prime fields $ \Bbb F_p $ (over a nonprincipal ultrafilter U) is a field of characteristic 0. How do I show that it has exactly one extension of degree n for each ...
3
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2answers
296 views

Elementary embedding and elementary equivalence [duplicate]

I guess it is not difficult, but I spent an hour thinking about this without success. Elementary embedding implies elementary equivalence, but elementary equivalence between structures does not ...
15
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1answer
297 views

Is $\Bbb R$ definable in $(\Bbb C,0,1,+,*,\exp)$?

Is there a first-order formula ϕ(x) with exactly one free variable $x$ in the language of fields together with the unary function symbol $\exp$ such that in the standard interpretation of this ...
5
votes
1answer
274 views

finiteness and first order sentences

Lets consider a set of sentences $T$ and a signature $\sigma$. I proved (using compactness theorem) that when $T$ has arbitrary large models than also an infinite model. Now there are several ...
5
votes
2answers
353 views

Meaning of quote: “model theory = algebraic geometry - fields”?

On the wikipedia article for model theory, it says that a modern definition of model theory is "model theory = algebraic geometry - fields" and cites Hodges, Wilfrid (1997). A shorter model theory. ...
3
votes
3answers
517 views

Complete theories - dense linear order

There are two things I would like to prove. DLO - Dense linear order is complete, that means that when $\psi$ is a sentence of the language $\{<\}$ then $DLO\vDash\psi$ or $DLO\vDash\neg\psi$ ...
4
votes
0answers
164 views

Formalizing model-theoretical large cardinals in a formal system for ZFC

I work with the Metamath formal system, which is expressive enough to define ZFC and prove some nontrivial stuff, but my current goal is to define large cardinals, and I'm hitting a wall somewhere ...
1
vote
1answer
57 views

Two questions on stable groups

I'm going over Marker's Model Theory. I have two questions in the "$\omega$-Stable groups" section. In Lemma 7.1.12: $p\in S_1(G)$ and $\psi(v)$ defines $G^0$. He claims that there exists a $b\in G$ ...
5
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1answer
229 views

In ZF, does there exist an ordinal of provably uncountable cofinality?

Question is in the title. In ZFC, one can prove that $\aleph_{\alpha+1}$ is regular, so there is a large source of cardinals with uncountable cofinality, but in ZF, it is consistent that ${\rm ...