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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90 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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45 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 ...
4
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2answers
53 views

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 ...
2
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1answer
67 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 ...
8
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1answer
113 views

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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1answer
99 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
40 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 ...
0
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2answers
58 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
28 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 ...
3
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0answers
66 views

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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0answers
49 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 ...
0
votes
1answer
50 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 ...
1
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1answer
43 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 ...
9
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1answer
120 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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66 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
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1answer
74 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
42 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 ...
2
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2answers
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 $S_1(A)\le\kappa$ ...
1
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1answer
27 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 (+, ...
5
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1answer
71 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
votes
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 ...
4
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0answers
57 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
143 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 ...
6
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2answers
184 views

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
27 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 ...
1
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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. ...
1
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2answers
77 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 ...
0
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2answers
81 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 ...
0
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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 ...
1
vote
1answer
49 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 ...
5
votes
2answers
87 views

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, ...
2
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1answer
45 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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0answers
39 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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0answers
55 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 $\infty$, ...
2
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1answer
62 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 ...
3
votes
1answer
73 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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0answers
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 ...
2
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0answers
43 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 ...
11
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6answers
718 views

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 ...
2
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0answers
58 views

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 ...
2
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1answer
66 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: ...
2
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1answer
92 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?
0
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2answers
68 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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1answer
26 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 ...
1
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1answer
23 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 ...
3
votes
1answer
100 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 ...
2
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1answer
61 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 ...
2
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1answer
37 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. ...
2
votes
1answer
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 ...