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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Limits of finite structures - first order logic

Assume that $\mathcal{C}=\{M_i:i\in I\}$ is an infinite collection of different finite $\mathcal{L}$-structures in a first-order language $\mathcal{L}$. The question is: What kind of infinite "...
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Lemma (?) on syntactically saturated maximal sets of sentences and existential sentences

Is there a proof of the following claim? $$\not \vdash_{{\rm FS}(L,M)} \exists x ~\alpha \implies [\alpha][x/c] \tag{T}$$ where no variable other than $x$ occurs free, $c$ is a name, ${\rm FS}$ ...
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126 views

What is the importance of “variety of algebras” in Universal Algebra?

Given an algebraic category, Birkhoff's Variety Theorem gives a categorical characterization of the full subcategories whose object-class forms a variety (i.e. can be defined by equations in the sense ...
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56 views

Complete theory with a infinite model has only infinite models

Let $T$ be a complete first order theory with a infinite model $M$. I want to show that every model $N$ of $T$ must be infinite. Since $T$ is complete then for every sentence $\phi$ we can either ...
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1answer
59 views

Algebraic numbers in real closed fields

I have been looking at the discussion of real closed fields in Appendix B of Marker's Model Theory:an Introduction. I am baffled by what it says about the uniqueness of real closures. I have no ...
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105 views

I don't really understand what a model is.

I've studied a bit of first order logic, and I still don't understand what a model really is. A model of a theory $T$ is an interpretation which assigns the value True to its sentences. Ok, that'...
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53 views

On definable bijections $b:M^n\rightarrow M^m$ in an o-minimal structure $\mathcal{M}$.

Let $\mathcal{M}=\{M,<,\ldots\}$ be an o-minimal first order structure, namely a structure where every definable set in $M$ is finite union of points and intervals with endpoints in $M\cup \{\pm\...
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34 views

Non-basic pseudo-elementary class

Is there a pseudo-elementary class, which is not an elementary class itself, of structures which is not basic pseudo-elementary, that is, is not the reduct of a basic elementary class?
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66 views

Real numbers with addition, multiplication, and a transcendental constant

Consider the structure $(\mathbb R, +, *, r)$, where $+$ is addition, $*$ is multiplication, and $r$ is a transcendental real number. Are the associative and commutative properties for addition and ...
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23 views

Prove $\{\neg\tau\}\cup\{\sigma_n:n\in\Bbb{N}\}$ has a model by compactness

$\sigma_n$ is the statement "There are at least $n$ elements in the domain" $\sigma_n:\exists x_1 \exists x_2 ... \exists x_n (\neg(x_1=x_2)\wedge\neg(x_1=x_3)\wedge...allPossiblePairs)$ $\tau$ is a ...
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42 views

“There are at least n elements in the domain” is entailed from f is not injective and f is surjective

I am working within a first order language with one unary function symbol $f$ and no other nonlogical symbols. I have written down sentences for 'f is not injective' and 'f is surjective denoted by $...
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92 views

Can type theory be viewed as an alternative to model theory?

While type theory certainly has traditionally been used for different purposes than model theory, as noted in this Philosophy SE post, I wonder to what extent type theory could model model theory ...
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48 views

Structure/Model for first order language

For simplicity lets say $L$ is a first order language with a single function symbol $f$ and no other nonlogical symbols What would a structure/model, $M$ say, for $L$ look like? I know $M$ has some ...
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2answers
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If $\cal{I}$ is a indiscernible sequence over $A$ then it is indiscernible over $acl(A)$

Let $\cal{I}=(b_i\mid i\in I)$ be an infinite indiscernible sequence over $A$. And let $acl(A)$ be the algebraic closure of $A$. (all in some structure) I am trying to show that $\cal{I}$ is also ...
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1answer
69 views

An example of a sentence $\sigma$ s.t. $\text{GL}_n(\mathbb{Q}(\sqrt{3})) \models \sigma$ and $\text{GL}_n(\mathbb{Q}(\sqrt{2})) \not\models \sigma$

A. I. Mal'cev proved the following remarkable result concerning the elementary equivalence of general linear groups: given fields $\mathbb{F}_{1}$ and $\mathbb{F}_{2}$ and natural numbers $m$ and $n \...
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Is it true that $\mathbb{C}(x) \equiv \mathbb{C}(x, y)$?

It is easily seen that any two consecutive entries in the tower of fields given below are not elementarily equivalent in the language of rings: $$\mathbb{Q} \subseteq \mathbb{Q}(\sqrt{2}) \subseteq \...
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Is it true that $\mathbb{F}_{1}^{\ast} \equiv \mathbb{F}_{2}^{\ast}$ implies $\mathbb{F}_{1} \equiv \mathbb{F}_{2}$?

Let $\mathbb{F}_{1}$ and $\mathbb{F}_{2}$ be fields, and let $\mathbb{F}_{1}^{\ast}$ and $\mathbb{F}_{2}^{\ast}$ denote the corresponding groups of units. If $\mathbb{F}_{1}$ and $\mathbb{F}_{2}$ are ...
7
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100 views

Is there a simple way of proving that $\text{GL}_n(R) \not\cong \text{GL}_m(R)$?

Letting $\mathbb{F}_{1}$ and $\mathbb{F}_{2}$ be fields, and letting $n \geq 3$ and $m$ be natural numbers, it is known that $\text{GL}_{m}(\mathbb{F}_{1})$ and $\text{GL}_{n}(\mathbb{F}_{2})$ are ...
2
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1answer
44 views

The cardinality of definable sets in RCF or RCF has no Vaughtian pair

Let $RCF=Th(\mathbb R ,+,\cdot ,<,0,1)$. We know RCF is $o$-minimal, i.e every definable set is a finite union of points and intervals with endpoints. I want to show that definable set in that ...
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2answers
67 views

Which of the following formulas are valid?

I have to find which of the following formulas is valid , and if it is not valid, give a model to show it. $\forall u(P(u)\rightarrow R(u))\rightarrow(\forall uP(u)\rightarrow \forall uR(u))$ $(\...
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1answer
35 views

Is every conservative extension an extension by definitions?

Suppose we have a theory T and a theory T' with corresponding languages L and L', where T' is a conservative extension of T. Must it be the case that T' proves that every relation symbol or constant ...
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Defining the Squaring Function in $(\mathbb{Z}; 0, 1, +, -, |)$?

I'm trying to show that the map $x\mapsto x^2$ is 0-definable in the structure $(\mathbb{Z}; 0, 1, +, -, |)$ (group of integers with the divisibility relation). But I'm not sure how to proceed. Any ...
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55 views

Showing the existence of a countable model for a nonstandard arithmetic

I apologize if the title of this post is in any way incorrect, please feel free to point it out. Let $\Gamma$ be the set of sentences of the language of arithmetic that are true in the standard ...
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1answer
53 views

If two sentences have the same enumerable models, then they are logically equivalent.

I am attempting to show that if two sentences have the same enumerable models, then they are logically equivalent. I am told that I need to apply the Löwenheim-Skolem theorem (if a set of sentences ...
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1answer
44 views

Properties of quantifiers on embeddings and interpretations

Let $\mathcal{M}_0$ and $\mathcal{M}_1$ be any interpretations. An embedding of $\mathcal{M}_0$ into $\mathcal{M}_1$ is a function $j:\mid\mathcal{M}_0\mid \rightarrow \mid \mathcal{M}_1\mid$ such ...
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1answer
122 views

Results in mathematics whose only proof is model theoretic

What are results in mathematics, for example in algebra, whose only proof so far used model theoretical arguments?
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68 views

Minimal model of ZF with $0\sharp$

We know that the constructible universe $L$ is an absolute and minimal model of ZF (every standard model of ZF contains "an" $L$, and it is actually the same $L$ for all of them). It is also my ...
3
votes
1answer
75 views

Equational identities of real multiplication augmented by a real number

Consider the structure $(\mathbb R, *, r)$, where $r$ is a real number that is neither $0$, $1$, or $-1$. Are the commutative and associative identities already sufficient to derive all the ...
2
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1answer
43 views

for a given $\kappa$, find a model of DLO with a greater cardinality that has a dense subset of size $\kappa$

So given a cardinal $\kappa$, i am trying to find a dense linear order with no end points of cardinality greater then $\kappa$ which has a dense subset of size $\kappa$ I think I'm almost there What ...
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61 views

RG is not $\kappa$-stable for all $\kappa$

I am trying to show for my home work that the theory of random graphs RG, is not $\kappa$-stable for every $\kappa$ i.e If $M\vDash RG$ and $A\subseteq |M|$ with $|A|\le\kappa$ then $S_1(A)\le\kappa$ ...
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1answer
30 views

Axiomatizable class of algebraic structures which is not the reduct of a variety

In the signature, (+), the class of groups is an axiomatizable class of algebraic structures which, though not a variety in that particular signature, is the reduct of a variety in the signature (+, 0,...
5
votes
1answer
74 views

Symmetry of Forking

In his paper: A Survey of Basic Stability Theory (http://link.springer.com/article/10.1007%2FBF02760649#page-1) Makki presents two (basic) facts: We work in a monster model of a complete stable ...
2
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1answer
65 views

Sufficient condition for $A \cong B \preccurlyeq C$ to entail $A \preccurlyeq C$

Let $A,B,C$ be models and suppose that $A \cong B \preccurlyeq C$. What simple additional requirement is sufficient to entail that $A \preccurlyeq C$? Notes: To see that this does not always follow,...
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59 views

Completing ordered Fields

How do these two forms of completion behave (in NBG) when fields are authorized to be proper classes? $(i)$: Every ordered field has a real closed, algebraic extension. $(ii$): Every ordered field ...
5
votes
1answer
145 views

The Reals are not interpretable in the complex numbers

Let $L=\{+,\dot{},0,1\}$ be the language of fields. I wish to show that the reals ($N=(\mathbb{R},+,\dot{},0,1)$) are not interpretable in the structure $M = (\mathbb{C},+,\dot{},0,1)$. I have the ...
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Are existentially defined subsets of affine algebraic sets unions of a finite number of affine algebraic sets?

Consider a set of polynomials in $\mathbb{C}[x_1,\dots,x_n]$. The zero locus of these polynomials $Z$ is a subset of $\mathbf{A}^n$ and is an affine algebraic set. Now, consider the following subset ...
2
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1answer
31 views

Show that a sequence of elements each realizing an isolated type over the previous realizes an isolated type

I'm trying to prove the following result which seems correct to me, but I'm not sure how to proceed. The proposition is: Let $M$ be a structure, $A\subseteq M$, $(a_1,\dots,a_n)$ be a sequence ...
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0answers
68 views

satisfaction of a sentence with two quantifiers

I want to be sure that I understand how to show that a structure satisfies a sentence under a variable assignment, and suspect that I'm handling the computation of multiple quantifiers incorrectly. ...
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2answers
87 views

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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2answers
93 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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1answer
35 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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1answer
52 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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1answer
52 views

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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42 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 $M_s$...
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66 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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1answer
71 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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1answer
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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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39 views

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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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 ...