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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Examples of Jónsson Models

Let $T$ be a complete first order theory. Suppose that $M\models T$. Then, $M$ is said to be a Jónsson Model of $T$ if for all $N$, such that $N\prec M$ and $N\models T$, we have $|N|<|M|$ (Note ...
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170 views

Every elementary submodel of $H(\aleph_1)$ is transitive

I am trying to solve the first part of Exercise II.17.30 in Kunen's Foundations of Mathematics which asks: Prove that every $A \preccurlyeq H(\aleph_1)$ is transitive. Here we are working over ...
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115 views

Ultraproducts of models of ZFC

Let $D$ be a non-principal ultrafilter over $\mathbb{N}$. Let $A_i$ for $i\in\mathbb{N}$ be (countable) models of ZFC such that $A_i\models \mathfrak{c}=\aleph_i$. Then, what is the size of the ...
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704 views

What is the real meaning of Hilbert's axiom of completeness

According to Greenberg's book of geometry it is sufficient to consider the axiom of Dedekind along with Hilbert's axioms (except of course for the Archimedian Principle and his Axiom of Completeness) ...
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93 views

The class of finite groups (models) and that of countable groups are not elementary classes (a generalized version).

First some definitions: For a set $\Sigma$ of $\mathcal{L}$-sentences, $Mod(\Sigma)$ denotes the class of all models that satisfy $\Sigma$. For a class $\mathcal{M}$ of models, we say it is $EC$ if ...
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87 views

Models of infinite groups and 'Group-like' objects

Let $G$ be an infinite group, and for simplicity, we will assume that $G$ is also countable. Now, with $G$ in mind, we construct a new language $L_G=\{f_{a_i\_},f_{\_a_i}:a_i\in G\}$ where ...
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512 views

There is no smallest infinitely large prime

I'm reading Kunen's Foundations of Mathematics and trying to solve Exercise II.16.19, which constructs an elementary extension of the reals as an ordered field, and asks the reader to prove various ...
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235 views

Universe cardinals and models for ZFC

I'm reading through Joel David Hamkins' set theory lecture notes. On page 14, on the subject of inaccessible cardinals and submodels of ZFC in $V$, he defines a universe cardinal to be a cardinal ...
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75 views

How can we get rid of function symbols in model theory?

I understand the presentation of a language in logic as having relations with arities, functions with arities and constants. I understand that a constant can be thought as a function with arity $0$. ...
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1answer
174 views

What is the point of (Compactness Theorem in the) Overspill Principle?

I am trying to understand the basics of computation theory. The Overspill Principle (also at google) basically says if you are cool you can do everything Let Г be a sentence of predicate logic ...
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1answer
144 views

Existence of an automorphism of $\Bbb C$ that fixes a finite set pointwise but does not fix $\Bbb R$ setwise

I would like to construct a (ring-theoretic) automorphism of $\Bbb C$ that fixes a finite set $A$ pointwise but does not fix $\Bbb R$ setwise. Marker's Model Theory, Corollary 1.3.6 does that in this ...
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50 views

Partial order version of elementary equivalence

Elementary equivalence is an important concept in mathematical logic. Two models $\mathfrak{M}$ and $\mathfrak{N}$ of the same signature are elementarily equivalent, written $\mathfrak{M} \equiv ...
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59 views

Does having a filter which is not maximal implies the negation of Łoś theorem?

If we have a family $(\mathfrak{M}_i)_{i\in I}$ of $L$-structures, and a filter $\mathcal{F}$ over $I$, we can define the reduced product $\prod_{i\in I}\mathfrak{M }_i/\mathcal{F}$. If $\mathcal{F}$ ...
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3answers
137 views

Are surjective polynomial maps injective?

An injective polynomial map $p:\mathbb{C}^n\mapsto\mathbb{C}^n$ is surjective (Ax-Grothendieck theorem). What is known about the reverse implication (surjective implies injective)? Why does the ...
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121 views

Showing $2$ is not definable in $(\mathbb{Q},+)$.

As stated, I'm to show that $2$ is not definable in $(\mathbb{Q},+)$. I tried proving it by contradiction by showing that if $2$ were definable, then we could define $\mathbb{N}$ and multiplication ...
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3answers
176 views

Goedel's completeness theorem in and/or for intuitionistic first order logic

Warning: I am neither a logician nor a set theorist, just curious about foundations of classical and intuitionistic mathematics. Therefore it might well be that the things to come are plain wrong, and ...
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48 views

Prove that Th(N, +, .) has uncountably many 1-types over some finite set

not sure how to go about answering the above question. Thanks for your help! (alternative: Prove that Th(N, +, .) has uncountably many n-types over the empty set.)
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315 views

Is it possible to prove the axiom of infinity from the real number axioms?

What I mean by the title is whether, given that there is a class $\Bbb R$ and operations $+,\cdot,<$ that satisfy the ordered field axioms and the least upper bound axiom, you can prove the ...
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107 views

Showing there does not exist a formal proof of a formula $\phi$.

My problem:Suppose $R$ is a binary predicate and use the soundness theorem to show that there does not exist a formal proof of $$ \phi =\forall y\exists xR(x,y)\rightarrow \exists x\forall yR(x,y).$$ ...
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2answers
309 views

Finite Model Theory

It seems that finite model theory is regarded (in a sense) as a computer theoretic subject. Is this the case or are there questions of interest that are of interest to mathematical logicians or more ...
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104 views

“Syntactic models” and the proof of the Completeness Theorem

In Computational Complexity by Papadimitriou (page 107), he outlines the basic idea for a proof of the completeness theorem for first-order logic - namely, that given a consistent set $\Delta$ of ...
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How is the Gödel's Completeness Theorem not a tautology?

As a physicist trying to understand the foundations of modern mathematics (in particular Model Theory) $-$ I have a hard time coping with the border between syntax and semantics. I believe a lot would ...
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221 views

Non-isomorpic uncountable dense linear orderings (a textbook example)

We know that two countable dense open orderings are isomorphic, and we know that this is generally not true for uncountable structures. Now I quote from the textbook, that is "Model-Theoretic Logics": ...
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Finding a finite model

Hello I am having difficulty with this question, I am not even sure what strategy one would go about proving something like this: Suppose $L$ is a language which includes an infinite list ...
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86 views

Is there a name for models whose every element is named by (one or more) variable-free terms?

Let $T$ denote a first-order theory. Is there a name for those models $M$ of $T$ such that for all $x \in M$, there is a variable-free term in the language of $T$ whose interpretation under $M$ is ...
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233 views

ZFC Axioms to be extended?

Sorry if this is going to be a really loaded question. I was told several times that for virtually all theorems/corollaries/propositions of mathematics (except those cases not compatible with ZFC ...
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247 views

Definable subsets of the natural numbers using only the successor function

Consider the first-order language whose only nonlogical symbol is the unary function symbol $S$, and the structure $\mathfrak{N} = ( \mathbb{N} , S )$, where $S$ denotes the successor function. Why ...
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49 views

First-Order Logic: Non-Normal Model of Sentences True in all Normal Models?

Let $\mathcal{L}$ be a first-order language with quantifier $\forall$, connectives $\neg$ and $\rightarrow$, a two-place predicate $E$ and a one-place function symbol $f$. There are no other ...
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361 views

Model-theory and Proof-theory in Propositional Logic

I'm trying to link results of model theory and proof-theory in propositional language. Here i will use $\models$ to denote logical consequence, in the model-theory sense. Being $x,y$ two formulas of ...
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2answers
99 views

Which constructions on a category are still interesting for a groupoid?

By a groupoid, I mean a (small) category in which every morphism is an isomorphism. It looks to me that constructions on a category like the opposite ("dual") category or products and (co-)limits ...
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209 views

Is a homomorphism expected to be a (structure-preserving) map?

Is a homomorphism a special type of morphism, namely a structure-preserving map? For a morphism (of a category), it is clear that we can't always expect that a morphism is necessarily a ...
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Putting down axioms for some symbols. Playing with their consequences qualitatively and symbolically. Building theories. The book?

I am interested in the design and building of theories. By building theories, I mean putting down axioms of various kinds, over various fields, exploring their perhaps interesting, or probably boring, ...
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97 views

Peano's theory of arithmetic and Gödel's 1st Incompleteness Theorem

Let $\mathcal{N}$ be Peano's 1st order theory of arithmetic and $\mathscr{A}$ it's standard model (which we assume exists). Infer from Gödel's 1st Incompleteness Theorem that there exists a closed ...
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44 views

Can I say something about the cadinality of this model?

Let $L$ be some first-order language. Suppose $A$ is existentially closed in $K$, a class of $L$-structures whose age is at most countable, and age($A$) is at most countable set . Can we say anything ...
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I would like some textbook recommendations for model theory

I am a 3rd year undergraduate math student and would like to study model theory. . I have some background with set theory, ordinals etc and also with mathematical logic. This is purely for self study ...
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208 views

Can all theorems of $\sf ZFC$ about the natural numbers be proven in $\sf ZF$?

I know a proof of Hindman's theorem that uses ultrafilters on the natural numbers, and ultimately, the axiom of choice. But the theorem itself is essentially a combinatorial property of the natural ...
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125 views

Example of two finitely isomorphic structures (or $\omega$-isomorphic) that are not partially isomorphic?

One can define several maps between given structures in order to portray similarities or differences. The concept of isomorphism displays a deep structural overlapping, while for instance an ...
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79 views

A Question About Tennebaum's Theorem?

Tennenbaum's theorem proves there are no countable recursive nonstandard models of Peano arithmetic. It is a proof by contradiction. If our countable, nonstandard model is recursive, then, given a ...
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1answer
93 views

Ax-Kochen theorem

I'm following the proof of Ax-Kochen theorem by this paper: http://arxiv.org/abs/1308.3897 I have two question in the proof of Ax-Kochen Principle: In the page 49, I didn't understend the ...
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2answers
226 views

An exercise on (isolated) types

I am currently working through a model theory course and am doing some exercises from Marker's book. I am currently attempting exercise 4.5.2. Which states the following: Let $T$ be the theory ...
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196 views

Non forking extensions of types as extensions of filters

Given a set of parameters $A$ a type in $S_n(A)$ may be thought of as a maximal filter on the monster model which can be constructed from $A$-definable subsets. Given a type $q\in S_n(B)$ saying that ...
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85 views

Show that every existential sentence is preserved upwards

A sentence is existential if it is of the form $\exists x_1 ... \exists x_nR$ and $R$ has no further quantifiers. A sentence is preserved upwards if and only if whenever it is true in an ...
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143 views

How are models constructed?

As I understand, a model $\langle S,\sim\rangle$ of $\mathsf{ZFC}$ is a set $S$ coupled with some binary operation $\sim$, in which the axioms of $\mathsf{ZFC}$ hold (please correct me if I am wrong). ...
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1answer
43 views

Prove $\{e\}$ is a model of group theory

In group theory one learns that there is exactly one trivial group of size 1, namely $\{e\}$. In addition to the axioms of the group theory, this group is uniquely determined by the axiom: DIM1 ...
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40 views

Definition of dicrete ordering

What means by "discrete ordering"?
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83 views

The definition of interpretation in a Kripke model collides with my intuition of what it should do

In Lindröm and Segerberg (2007) exposition of a Kripke model, with frame $F= \langle W,D,R,E,w_0\rangle$, they define an interpretation $I$ as a family of functions $I_w$, where $w$ ranges over ...
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71 views

Are all theorems of minimal arithmetic theorems of a given theory?

I am working on some metamathematics revision and the following question came up. Let the theory $R_0$ be axiomatized by the following axiom schemata which hold for all $n,m \in \mathbb{N}$: ...
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107 views

The unique model of cardinal $\kappa$ of a $\kappa$-categorical countable theory is saturated.

Let $T$ be a $\kappa$-categorical ($\kappa \geq \aleph_1$) first-order theory in a countable language $\mathcal L$. I try to prove that its unique (up to isomorphism) model of cardinal $\kappa$ is ...
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266 views

What is the definition of “Winning Strategy” in an Ehrenfeucht-Fraïssé game?

I've read many descriptions and applications of a Winning Strategy, as much as many for a Strategy tout court, but when a formal, algebraic definition is called upon, I've found close to no input. I ...
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1answer
339 views

What is the use of Tarski-Vaught test?

As title says, what is the use of Tarski-Vaught test? I do understand that it is necessary and sufficient criteria for $N$ to be an elementary substructure of $M$, but beside that I don't see how this ...