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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Existance of an (in)finite theory having infinite model

Please help me to study the following simple cases: Let $P$ be a binary predicate symbol. I am trying to find out, if there exists a satisfiable $T$ having infinite models only, for the following ...
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30 views

Is it possible to have logic without syntax (with only semantic proof methods)?

In one paper I have read a note "Thus, unlike approaches which make use of full first order logic, unprovability of a formulae with respect to a agent specification can be shown by each of two ...
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30 views

Embeddable rings axiomatic class?

In this question, a ring is defined to be with a unit distinct from the zero element, not necessarily a commutative ring though. Is the class of all such rings that can be embedded into fields an ...
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26 views

First-order logic: largest size among smallest finite models for formulas of a given length

Apologies for the somewhat cryptic title. For any first-order formula X, let ssm(X) be the size of the smallest finite model of X. By size I mean number of individuals. So, for example, ssm('Fa ...
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Is there a first order formula $\varphi[x]$ in $(\mathbb Q, +, \cdot, 0)$ such that $x≥0$ iff $\varphi[x]$?

In the first-order language $\mathscr L$ having $(+, \cdot, 0)$ as signature, it is easy to define a formula $\phi[x]$, namely $\exists y \; x = y^2$, satisfying : $$\text{for all } x \in \Bbb R, ...
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algebraic closure is the intersection of all elementary sub-models of the monster

This is a question from an exercise in model theory. Let T be a complete theory, $ \mathfrak{C} $ monster model of T (a $ \kappa $ saturated model of cardinality $ \kappa $ for some large $ \kappa $) ...
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61 views

Why do ultraproduct structures use a quotient as their universe?

For an $L$-structure $\mathfrak{A}$ with universe $A$, if we have an index set $I$, with an ultrafilter $U$, we create an ultraproduct structure having as its universe $\Pi_I \;A_i/U$. This is the set ...
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35 views

Proof that the inverse limit of an inverse system is equal to another set

I'm trying to to learn model theory and so working with some basic examples. Consider the following: Let $D$ be finite subsets of $\mathbb{Q}$ with the ordering given by the subset relation. Let ...
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48 views

Theories of Arbitrary Morley Rank

Suppose that you have a language $L$. I can show that theories like DLO, or any unstable theory for that matter, has Morley Rank $\infty$. I can also show that $REI_\alpha$ has Morley rank $\infty$, ...
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Is Induction Independent of the Other Axioms of PA?

I am trying to come up with a model of first order Peano Arithmetic (PA) where induction fails. Let $PA^{-IND}$ have the same axioms as PA except the first order induction axiom schema is replaced ...
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Does a finite first-order theory which has a model always have a finite model?

I'm curious whether this is true or not: Let T be a finite, first-order theory. If T has a model, then T has a finite model. I would assume the answer is 'yes', but I wanted to make sure I ...
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45 views

$\omega$-categoricity and infinite languages

The Ryll-Nardzewski Theorem states that an equivalent condition to $\omega$-categoricity is that there is a finite number of $n$-types for any $n$. So what happens when you add a countably many unary ...
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24 views

Is there a conservative extension of IZF that extends IZF by a weak form of the axiom of choice?

The full axiom of choice implies the LEM, and so is incompatible with constructive mathematics, although there are some weaker forms of the axiom of choice, such as the axiom of dependent choice, or ...
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131 views

On the Decision Problem for Two-variable First-Order Logic

I have a question concerning the model construction of the $\forall \forall \land \forall \exists$ - Scott sentence on page 6 in this paper: www.cs.rice.edu/~vardi/papers/basl96.ps.gz Why do we ...
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165 views

Does this proposition hold if $\text{Mod}(\Gamma)=\emptyset$?

The following is the start of basic corollary in my logic text: For any set $\Gamma$ of sentences, $\Gamma\subseteq\text{Th}(\text{Mod}(\Gamma))$. What happens when $\text{Mod}(\Gamma)$ is empty? ...
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Are ($Q$, $\leq$) and ($Q \times Q$, $\leq _e$) isomorphic? [duplicate]

I can't really tell if ($Q$, $\leq$)$\cong$($Q \times Q$, $\leq _e$), where $\leq_e$ denotes the left lexicographic order. Neither have a last/first element, both are dense and have the same ...
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Definibility of $\mathbb{Z}$ in product rings

If $R$ is a product ring whose factors are in a finite number and are all quotients of $\mathbb{Z}$ (that is, either $\mathbb{Z}$ or $\mathbb{Z}_n$'s ), is it a sufficient and necessary condition for ...
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33 views

Extending the language in Henkin style completeness proof for first-order logic

There is a detail in the Henkin style proof of completeness for first order logic that I can't quite understand. So in the first part (Lindenbaum's Lemma), we need to show that a consistent set of ...
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85 views

Is $\mathbb Z$ first-order definable in (the ring) $\mathbb{Z\times Z}$?

Is $\mathbb Z$ first-order definable in $\mathbb{Z\times Z}$ (using sum and product but obviously not the concept of "component")? I believe no but how may I prove it? Is this standard?
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155 views

Can't EF game theory be applied to finite languages WITH function symbols?

Let $\mathcal{M}$ and $\mathcal{N}$ be two structures in a language $\mathcal{L}$. We define the finite determined game $G_n(\mathcal{M},\mathcal{N})$ as a game with $n$ rounds where in each round ...
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63 views

Boolean model containing both confusion and junk

I'm doing a course in Equational Programming, and really new to these materials. So we got a specification for Booleans: ...
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65 views

Is it possible to formalize $T \models \sigma$ within the deductive calculus?

I know that the notion of $T \vdash \sigma$ is formalizable within any sufficiently powerful theory $T$ but is $T \models \sigma$ formalizable as well? How is this possible if there are infinitely ...
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22 views

On the existence of finite substructures when sufficient chain conditions are met

Let $L$ be a language and $T$ and $L$ theory. Suppose that for any $M\models{T}$, we have $M\subseteq{\bigcup{C_{n}}}$, where each $C_{n}\models{T_{\forall}}$ is finite. I want to show that for some ...
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19 views

1-model complete

For $L$ structures $A$ and $B$ we write $A\preceq_{1}B$ if $A\subseteq{B}$ and $A\models{\varphi(a)}$ iff $B\models{\varphi(a)}$ for any finite tuple (of the correct length) $a$ from $A$ and for any ...
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81 views

Class models of $\mathsf{ZFC}$ and consistency results

First of all, I'm only starting to study independence results in set theory. And there is one obstacle that confuses me a lot. Probably such questions have already been asked, but I haven't found ...
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1answer
46 views

Indiscernibles over a model

Working within the framework of a monster model, I wish to show that: (*) If $(a_{i}:i<{\lambda})$ is an indiscernible sequence over $A$, then there is a model $M$ containing $A$ such that ...
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54 views

D.Marker's axiomatization of rings

Adding "-" as a binary function to the language of rings and the sentence $∀x(x+(−x)=0)∀x(x+(−x)=0)$ to the set of axioms proves existence of additive inverses. But I can't see how Professor Marker's ...
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29 views

Definition of Local Finiteness

Let $L$ be a language and let $T$ be an $L$-Theory. $M\models{T}$ is said to be locally finite if for any given finite subset $X$ of $M$, there is a finite substructure $A$ of $M$ s.t. ...
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1answer
48 views

A test for quantifier eliumination

In David Marker's "Model Theory: An Introduction" book I was trying to prove Corollary 3.1.12 which is left for the reader, but I couldn't reach any solution. The aforementioned corollary states: ...
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54 views

An AE axiomatization of groups

Let $L=\{*\}$. The usual axiomatization of groups in this language has the EA axiom $\exists{e}\forall{x}$ $ e*x = x$. But the union of a chain of groups is also a group. This means that the theory of ...
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66 views

Upward Löwenheim Skolem

I'm trying to understand the proof of (a version) of the upward Löwenheim Skolem Theorem, which states that given a language $\mathscr{L}$ and a set of $\mathscr{L}$-sentences $\Sigma$ with a ...
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1answer
53 views

elementary equivalence and incompleteness

I read the following line in a text on set theory: "Peano Arithmetic has continuum many non-isomorphic countable models (including the standard model omega), all of them elementary equivalent." ...
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1answer
62 views

ACF universal is the theory of integral domains

When studying David Marker's "Model Theory: An Introduction" book trying to understand the proof of Lemma 3.2.1 which says: $ACF_{\forall}$ is the theory of integral domains, I couldn't understand the ...
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66 views

Absoluteness and Extensionality

In the set theory text that I am reading, the author writes: Relative to the set $A = \{ 0, \{\{0\}\} \}$, the sets $0$ and $\{\{0\}\}$ are indistinguishable in the sense that $[$for all $x$ in ...
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1answer
87 views

Definition for non-dividing

The definition for non-dividing is taken as the negation of the definition for dividing (as found in http://www.math.cmu.edu/~rami/simple.pdf : Definition 1.1 for example). Thus assuming ...
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98 views

Explicit countable elementary extension of $\mathbb{N}$

I would like to see an explicit example of a non-trivial elementary extension of the structure $(\mathbb{N}, +, \cdot, 0, 1)$ where $\mathbb{N}$ includes zero. Most of all I am interested in countable ...
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275 views

What's an example of a number that is neither rational nor irrational?

Of course in regular logic, the answer is there aren't any. But in intuitionistic logic, there might be, as seen by this answer: http://math.stackexchange.com/a/1437130/49592. My question is, as per ...
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Clarify definitions of relation and 0-ary relation

From mathworld.wolfram.com: A relation is any subset of a Cartesian product But if so, then the null set is all of: 0-ary, 1-ary, 2-ary etc. Wouldn't it be better to define it as: A relation ...
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Shelah's materialize vs. realize;note: tags are badly chosen due to the lack of them

Can someone please explain to me in some detail the exact difference between materialize and realize for a Galois type $p$? Esp. is realize a special case for materialize? Why is it so? What is the ...
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178 views

A Characterization of Categories with a Conservative Forgetful Functor to SET

Examples of categories over $\bf{Set}$ such that the forgetful functor is conservative include the "algebraic" categories of groups, rings, modules, monoids, etc., but does not include the ...
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When can independence of a statement in a theory be reduced to “truth”?

Since the Goldbach conjecture is in $\Pi_1^0$, if it were proven to be independent of Peano Arithmetic, it would follow that the Goldbach conjecture is true (i.e. true in the standard model), since ...
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88 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 ...
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1answer
39 views

Prove that the Morley Rank is preserved under definable bijections.

I need to prove this: If there is a definable bijection between $\varphi(C)$ and $\psi(C)$ then $RM(\varphi)= RM(\psi)$. Where $C$ is the monster model. I can intuitively understand it, the Morley ...
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Why can't an $\omega$-stable theory have finitely many countable models?

This is a "well-known fact," but I'm at a loss to finding a proof. I could swear I've read it somewhere, but checking the handful of places I'm used to checking doesn't help. Google gives nothing, ...
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Understanding types and the proof that every type is realized in an elementary extension.

So I've been recently been studying types from David Marker's book and have some issues understanding them and in particular why did Marker choose to present certain proof of the following theorem ...
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43 views

prove that any two isomorphic structures are elementarily equivalent

Imagine we have two L-structures $M$ and $N$. For each L-sentence $\phi$ , $M$ models $\phi$ iff $N$ models $\phi$. We call $M$ and $N$ two elementary equivalent L-structures. We say $M$ and $N$ ...
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1answer
68 views

Type of Infinite Tuple

Things along the following lines is often said about infinite tuples in model theory (we are assuming that we are working inside some monster model $M$ of some complete $L$ theory $T$): If I is a ...
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278 views

how to create the set of hyperreal numbers using ultraproduct

As title says, can anyone explain how to create the set of hyperreal numbers using ultraproduct from the set of real umbers?
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1answer
42 views

Forking in Strongly Minimal Theories

I have been trying to define $A\overset{\vert}{\smile}_{C}B$ in a strongly minimal theory (let's say countable to avoid complications though I'm not sure if this matters). My attempt is based on the ...
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Intervals in divisible ordered groups

Is it true that if $(G,+,0,<)$ is a divisible ordered abelian group with at least two elements, then for $a,b >0 \in G$, there is an injective order preserving map from $[0;a)$ to $[0;b)$? It ...