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 ...

learn more… | top users | synonyms

3
votes
2answers
26 views

Properties of an elementary substructure

Let $M$ and $N$ be structures for a first order language $L$, with $M$ an elementary substructure of $N$. This means that $M$ is a substructure of $N$ and if $\varphi(x_1,\ldots,x_n)$ is a formula ...
1
vote
1answer
56 views

What is a universal function in model theory?

What does it mean that a function in a model is universal? Let A be the domain of a model. As I understand it, an empty function is a function that is not defined for any object in A; an empty n-ary ...
0
votes
2answers
63 views

Does Second-Order Comprehension make second-order ZFC inconsistent due to Russell's Paradox?

When we do set theory, we take our first-order variables to range over all sets. But if we take our second-order variables to range over sets of sets in the range of the first-order variables, then ...
3
votes
1answer
40 views

Rigid relations and Choice

A binary relation $R$ on a set $D$ is rigid iff the unique $D → D$ bijection that fixes $R$ is the identity function. Any well-ordering is rigid, so the Well-Ordering Principle has the consequence ...
3
votes
1answer
70 views

Does no non-standard model of Peano Arithmetic make the integers a principal ideal domain?

Though I do not find a reference now, I have heard no non-standard model of Peano Arithmetic has a principal ideal domain as its ring of integers. Is that right? Is it trivial? Or is there a good ...
1
vote
1answer
60 views

Completeness Theorem in logic and Completeness of a theory

Completeness Theorem says: $\Gamma \models \phi \longrightarrow \Gamma \vdash \phi$ And from definition of satisfaction: $\neg(\Gamma \models \phi) \longleftrightarrow \Gamma \models \neg\phi$ Now ...
6
votes
2answers
90 views

Is the axiom of induction constructively verifiable for a non-standard model of Peano arithmetic?

There exist models of the natural numbers which include infinite numbers. Such models are called non-standard models of arithmetic. (Proof: by the compactness theorem, there exist models of Peano ...
1
vote
1answer
33 views

Model isomorphisms of a set of sentences

I have a question about models of a set of sentences $T$, specifically the following: Let $S=\{R\}$ where $R$ is a unary relation symbol. Let $T$ be the set of sentences that for each $n\geq 1$ ...
3
votes
2answers
46 views

Prove that there exists a sentence $\varphi$

I'm doing some excercises from the book "The Incompleteness Phenomenom" from Goldstern and Judah. I have to do this and I don't know how to start. Let $\Sigma_1 $ and $\Sigma_2$ be sets of sentences ...
4
votes
3answers
71 views

Undefinability of evenness in first order logic

My question is to show there is no sentence $\psi$ in a language of first order logic without any non-logical symbols such that for every finite structure $\mathcal{A}$: $$\mathcal{A} \vDash \psi \; ...
4
votes
3answers
105 views

Minimal model of ZFC without power set axiom

We know that $L$ is the minimal standard model of ZFC. The question is, what is the minimal "standard" model of ZFC$^-$ (meaning ZFC without the Power Set axiom)? This is really two questions: Is ...
4
votes
2answers
74 views

Non existence of Prime models

Let $L$ be a countable language. Let $T$ be a complete $L$ theory. We know that if $T$ is small, then there is a prime model of the theory. But $\text{Th}(\mathbb{N},+,\times,0,1)$ is not small but it ...
1
vote
2answers
50 views

elementary question about definability

My understanding is that an object, m, from the domain of a model, M, is definable by a formula, F(x), just in case M |= (Vx)[F(x) <----> x = m]. However, this assumes that there is a name for the ...
2
votes
3answers
60 views

About definition of model

In Model theory, the definition of a model is a set. Can it be a proper class? ZFC has a model and maybe some models is a proper class. Definition of a model needs to include a proper class. Is it ...
0
votes
0answers
84 views

Gödel's incompleteness theorem applys to ZFC theory

When I assume ZFC's consistency, it is impossible to prove ZFC's consistency in itself from Gödel's incompleteness theorem 2. If ZFC's consistency have done, its proof need to be done in stronger ...
2
votes
0answers
82 views

How can I understand about ZFC and Gödel's Completeness theorem [closed]

English 1 ZFC could be formulated as First order logic. 2 Gödel's Completeness theorem is a theorem within ZFC. 3 I think a lot of books about set theory is implicitly assuming Gödel's ...
1
vote
3answers
91 views

In relatively simple words: What could be a model of $\sf {ZFC}$?

An $\text{interpretation}$ of a theory consists of A domain of discourse $\mathcal U$, usually required to be non-empty. For every constant symbol, an element of $\mathcal U$ as its ...
1
vote
1answer
103 views

absoluteness and and transitivity

I'm early in my reading about absoluteness, but one thing has me stuck, so I thought I'd ask. One reason absoluteness seems to matter is that we feel confident that we know what we're talking about ...
1
vote
0answers
35 views

Affine space over $\mathbb {Q}$ prime model, and dimension.

For homework, i need to prove the following: Let $M$ be a structure with universe $\mathbb {Q}$ of the language that cosiest only of function symbols $f_{\bar {\alpha}}(\bar{x})$ for all $\bar ...
5
votes
1answer
116 views

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 ...
0
votes
0answers
25 views

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, ${\rm FS}$ ...
7
votes
3answers
118 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 ...
2
votes
2answers
49 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 ...
3
votes
1answer
56 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 ...
5
votes
1answer
101 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, ...
7
votes
1answer
52 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 ...
2
votes
0answers
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?
2
votes
1answer
65 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 ...
1
vote
1answer
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 ...
1
vote
0answers
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 ...
2
votes
1answer
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 ...
0
votes
1answer
46 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
votes
2answers
54 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
votes
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 ...
15
votes
1answer
131 views

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
votes
1answer
115 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
votes
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
votes
1answer
41 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
votes
2answers
62 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. $\forall u(P(u)\rightarrow R(u))\rightarrow(\forall uP(u)\rightarrow \forall uR(u))$ ...
1
vote
1answer
30 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
votes
0answers
67 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 ...
1
vote
0answers
50 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
51 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
vote
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
votes
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?
1
vote
0answers
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
votes
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
votes
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
votes
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
vote
1answer
28 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 (+, ...