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

2
votes
1answer
16 views

Elementary equivalence of models

I'm quite new to model theory, so please correct me if I'm using wrong terminology. I need help with an exercise from Smirnov's book "Varieties of algebras" (In Russian). Problem: Assume that a ...
4
votes
1answer
55 views

Decision and the Uncountable Spectrum

In 2000, Hart, Hrushovski, and Laskowski classified all complete first order theories in a countable language up to their uncountable spectra. However, does this also imply that given a $any$ ...
1
vote
1answer
29 views

Relative Interpretations alla Kunen

at the moment I try to figure out some details of Kunen's "Relative Interpretation" Definition (within the 2013 Edition of his "Set Theory", p. 99 to 100): Definition If $\Lambda$ is some axioms ...
1
vote
1answer
53 views

Are algebraic structures required to satisfy axioms?

Is it a requirement for algebraic structures, when studying universal algebra, to satisfy axioms? The reason I ask is because algebraic structures are only defined by a underlying set, a signature, ...
1
vote
2answers
256 views

Axiom Systems and Formal Systems

I'm a really beginner in Mathematical Logic.I'm currently reading Shoenfield Mathematical's Logic and i'm having a hard time trying to relate the concept of Formal Systems with the concept of Axiom ...
-2
votes
0answers
58 views

Is the structure with sets and the ZFC axioms a model of the first order logic?

Wikipedia says ZFC is a one-sorted theory in first-order logic. The signature has equality and a single primitive binary relation, set membership, which is usually denoted ∈. The formula a ∈ b ...
1
vote
1answer
72 views

Is for every ultrahomogenous structure M the theory Th(M) model complete?

A structure M is ultrahomogenous if every isomorphism between finitely generated substructures of M can be extended to an automorphism of M. A theory is model complete if every embedding between ...
22
votes
1answer
493 views

Model existence for infinitary logics

One of the problems of infinitary logic is that it is possible for compactness to fail in a spectacular way: for example, one can concoct an inconsistent set of axioms whose proper subsets are all ...
6
votes
2answers
168 views

Model theory in group theory

I am interested in useful results for group theorists that can be shown using model theory. For example : Theorem: Let $\langle X \mid R \rangle$ be presentation of a group $G$ with $X$ finite and ...
-1
votes
0answers
34 views

Is an o-minimal structure equivalent to a totally ordered set?

Is the notion of o-minimality synonymous to a totally ordered set? Both notions seem to emanate from Tarski although he may not have discussed o-minimality explicitly...
2
votes
0answers
70 views

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 ...
1
vote
2answers
62 views

An Uncountable language , A Model of $\mathbb{N}$, A Problem.

Edit 1: I messed up my original question, but Arthur Fischer answered my question anyway. Edit 2: We can actually restrict $L$ to the language in arithmetic with the prdicate $P_{\mathbb{P}}$. ...
6
votes
1answer
101 views

Elementary equivalence of free groups

This must be known inside out by model theorists by I have no cluse whether the following is true or not: Denote by $F_n$ the free group on $n$ generators. Suppose that $n\neq m$. Are the groups ...
3
votes
2answers
102 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 ...
2
votes
3answers
224 views

Upper bound on cardinality of a field

Is there an upper bound on the cardinality of a field? The "biggest" fields I know are the field of real numbers, or the field of complex numbers. Is there a field with cardinality greater than ...
5
votes
1answer
112 views

Which logic is stronger? SOL or $\frak{L}_{\infty,\infty}$?

Both second-order logic($\mathsf{SOL}$) and infinitary first-order logic $\frak{L}_{\infty,\infty}$ are proper extensions of first-order logic($\mathsf{FOL}$), that is, are extensions of a ...
5
votes
1answer
65 views

Is infinitary logics $\mathcal{L}_{\infty\omega}$ an abstract logic?

Infinitary logics $\mathcal{L}_{\infty\omega}$ is an extension of first-order logics such that $\bigvee\Phi \in \mathcal{L}_{\infty\omega}$ if $\Phi$ is a set of ...
6
votes
1answer
77 views

Construction of Ultrafilters

I've been doing a lot of work with ultrapowers and saturation recently. In particular, I am reading chapter 6 of Chang and Keisler as well as Keisler's paper on "Ultraproducts which are not ...
1
vote
1answer
27 views

Is “constructible from” a transitive relation?

In Jech's Set Theory, exercise 13.27, it is hinted that $X \in L[Y]$ and $Y \in L[X]$ together imply $L[X]=L[Y]$. I tried to prove this fact without success, although I suspect the proof is simple. ...
2
votes
0answers
97 views

How large or small can the gap in Shelah's main gap theorem be, up to consistency?

Shelah's main gap theorem in model theory says: For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of ...
1
vote
1answer
36 views

Is there an algebraic ω-categorical structure with quantifier eli., that is not ultrahomogenous?

The following result holds for relational structures: If $A$ is a countable structure, with an $\omega$−categorical theory $Th(A)$, that admits quantifier elimination, then $A$ is ultrahomogeneous. ...
2
votes
1answer
50 views

Why is $\alpha \mapsto L_{\alpha}[A]$ $\Delta_{1}$?

On page 187 of Jech's Set Theory, there is a proof sketch of the fact that $\alpha \mapsto L_{\alpha}$ is $\Delta_{1}$. As far as I can tell, Jech's argument only shows that this operation is ...
0
votes
0answers
37 views

Sets Constructible Relative To A Unary Predicate

The class $L$ of constructible sets is defined by recursion using the operation def$(M)=\{x \subset M: x$ is definable over $(M, \in) \}$. By adding a unary predicate, $P$, to our language, we can ...
14
votes
1answer
304 views

What lessons have mathematicians drawn from the existence of non-standard models?

So, as someone whose knowledge of mathematics has always come from studying it with an eye towards philosophical/foundational issues and studying it with other philosophers (who are not primarily ...
7
votes
2answers
184 views

Is the completeness theorem for first-order logic relative to one's choice of set theory?

By the completeness theorem for first-order logic, every consistent theory has a model. However, to even make sense of the word "model," I believe we're assuming a set theory. So is there a set theory ...
0
votes
0answers
36 views

Model-theory : questions regarding back-and-forth sets

See my previous post for the basic definitions from Jouko Väänänen, Models and Games (2011), page 54-on. See page 64 for : Definition 5.14 Suppose $\mathcal A$ and $\mathcal B$ are ...
10
votes
0answers
243 views

Number of automorphisms of saturated models

I have the following assignment question: Let $M$ be an $L$-model of cardinality $\kappa$. Assume $M$ is saturated. How can you show that $|\text{Aut}(M)|=2^{|M|}$? I see two possible ...
1
vote
1answer
45 views

Model-theory : questions regarding partial isomorphism

I'm having problems with the first pages of Bruno Poizat, A Course in Model: Theory An Introduction to Contemporary Mathematical Logic (ed or 1985), specifically with local isomorphism and back- and ...
2
votes
1answer
103 views

Cardinalities of Collections of Models

Let $T$ be a complete theory in a countable language (with only infinite models). Recall the spectrum function: $I(\aleph_\alpha,T)=$ the number of non-isomorphic models of $T$ of cardinality ...
2
votes
1answer
46 views

Does the countability of the structure matter for the connection between quantifier elimination, $\omega$-categorical and ultrahomogenous?

A relational structure $A$ with an $\omega-$categorical theory $Th(A)$ is ultrahomogenous iff $Th(A)$ admits quantifier elimination. I was wondering wether the structure $A$ has to be countable... ...
7
votes
1answer
150 views

Gödel's way of teaching non-standard models to Takeuti.

In Memoirs of a Proof Theorist, Gaisi Takeuti relates how Gödel taught him about nonstandard models in an "interesting" way: It went as follows. Let T be a theory with a nonstandard model. By ...
1
vote
1answer
70 views

Show that if $L$ is countable and contains a two-place predicate symbol, there are $2^{2^{\aleph_0}}$ classes of $L$-structures closed under $\equiv$

We say that a class of structures $K$ is closed under elementary equivalence ($\equiv$) if for all $A, B$, if $A \in K$ and $A \equiv B$, then $B \in K$. How to show that if $L$ (as a set of specific ...
4
votes
1answer
71 views

Model of complete extension of Zermelo set theory

Chang and Keisler's Model theory gives the following exercise problem: Prove that there is a complete extension $T$ of Zermelo set theory which has arbitrary large natural models. (A model ...
0
votes
1answer
52 views

Why are there $\leq$ $2^{\operatorname{card}(\operatorname{Form}(L))}$ elementarily nonequivalent structures for $L$?

Let $L$ be a set of specific symbols and $\operatorname{Form}(S)$ be the set of all first-order formulas over $L$. Why are there $\leq$ $2^{\operatorname{card}(\operatorname{Form}(L))}$ ...
2
votes
1answer
56 views

How to prove that $max(\aleph_{0}, card(X)) = max(\aleph_{0}, card(L(X)))$?

I struggle with the following problem. Let $X$ be a set of elementary sentences and $L(X)$ be the smallest elementary language in which we can express all the sentences from $X$. How to prove that ...
4
votes
1answer
46 views

A model which has only one undefinable element over a language with only a finite number of symbols

I try to solve the problem 1.3.14 in Chang and Keisler's Model theory: For each $n\in\omega$, find a model $\mathfrak{A}_n$ for $\mathcal{L}$ a language with only a finite number of symbols, which ...
6
votes
3answers
129 views

Can $(\Bbb N,\leq)$ have an $\aleph_0$-categorical theory (in a larger language)?

One of the nicer consequences of compactness is that there is no statement in first-order logic whose content "$\leq$ is a well-order". So we can show that there are countable structure $(M,\leq)$ ...
126
votes
1answer
3k views

Is there a 0-1 law for the theory of groups?

For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } ...
2
votes
0answers
37 views

Counter example of monotone union [duplicate]

I saw this exercise in "Elements of Abstract and Linear Algebra" by E. H. Connell: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, ...
1
vote
1answer
70 views

On the number of countable models of complete theories of models of ZFC

Fix the language of set theory $\mathcal{L}=\{\in\}$. Let $\langle M,\in\rangle$ be a set or proper class model of ZFC (e.g. $M$ could be $L$, $HOD$, $V_{\kappa}$ for some inaccessible cardinal ...
3
votes
2answers
96 views

Ultrapower and hyperreals

The construction I've seen of the field of hyperreal numbers considers a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, then takes the quotient of $\mathbb{R}^{\mathbb{N}}$ by equivalence ...
0
votes
1answer
33 views

Why is the cardinality of a language defined as $||\mathcal{L}||$?

I'm reading Chang and Keisler's Model Theory and I don't quite understand the notation they use for the cardinality of a language. Elsewhere in the book, the cardinality of a set $X$ is denoted by ...
2
votes
3answers
78 views

One-element model of first-order PA

The First-Order axiomatisation of PA is: $\forall x. x = x$ $\forall x, y. x = y \rightarrow y = x$ $\forall x, y, z. x = y \land y = z \rightarrow x = z$ $\forall x. 0 \ne S(x)$ $\forall x, y. S(x) ...
2
votes
1answer
77 views

Inuition regarding Lowenheim-Skolem applied to models of set theory

According to wikipedia, ...the Löwenheim–Skolem Theorem states that for every signature $σ$, every infinite $σ$-structure $M$ and every infinite cardinal number $κ ≥ |σ|$, there is a ...
8
votes
3answers
482 views

Relationship between Category theory and Axiomatic set theory

I've recently started learning Category theory- and I have a pondering- wondering if anyone can help. Is it possible for two categories to satisfy two different set-axiom system. Namely- is it ...
0
votes
1answer
35 views

elementary class and abstract elementary class

I think I confused with the concept of elementary class and abstract elementary class. We see in the definition of AEC that each elementary class is an AEC. Let $l=\{\le\}$, $T=\emptyset$, ...
3
votes
0answers
82 views

Countable transitive $T \vDash ZFC-P$ with $\approx$ not absolute: why do we need $H(\aleph_3)$?

I am trying to solve Exercise IV.3.31 from Kunen's Foundations of Mathematics. I think I have a solution but I am confused by one of the hints. By request, here is the text of the exercise. ...
3
votes
1answer
38 views

Existence of theories with exactly two countable models

I read a result of Vaught(a little down the page) that says that there cannot be any first order theory which has exactly two countable models upto isomorphism. Is this not a counter example: The ...
2
votes
1answer
64 views

Absolute confusion! (A question about absolute *sentences*)

I'm seriously confused about absoluteness. A formula in the language of a theory $T$ is absolute for $T$ structures if its truth value is the same in all standard transitive models of $T$ (this may ...
4
votes
1answer
84 views

Elementary equivalence versus equivalence between the total theory in model theory

In the page for elementary equivalence on wikipedia, in the introduction, they say: "If N is a substructure of M, one often needs a stronger condition. In this case N is called an elementary ...