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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87 views

Non-well-founded models viewing well-founded models as non-well-founded.

I'm currently thinking about how different models of set theory view each other. In particular I'm looking at how well-foundedness behaves between different models. So we have the Axiom of ...
2
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2answers
55 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 |= to denote logical consequence , in the model-theory sense. Being x,y two formulas of a ...
12
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3answers
288 views

Does every complete theory admit quantifier elimination?

Does every complete theory admit quantifier elimination? I know that at least in some simple cases the reverse is true; such as some reducts of number theory.Thanks
2
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1answer
63 views

Why Quantifier Free Formulas define Linear Functions.

How do you prove that functions definable by quantifier-free formulas must be linear? I am interested for the structure $\langle Q,+,0\rangle$.
2
votes
2answers
66 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 ...
3
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1answer
77 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 ...
5
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1answer
148 views

Why is there apparently no general notion of structure-homomorphism?

In model theory, one typically defines only embeddings of structures and isomorphisms, but I haven't seen a definition of general structure homomorphisms. Is there some particular reason behind that? ...
3
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1answer
79 views

construction set of natural number logic

I identify the natural number $0$ with the empty set $\emptyset$, $1$ with $S(0)$, $2$ with $S(1)$, etc, etc. The axiom of infinity says $\exists x (\emptyset\in x\wedge \forall z\in x\space ...
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2answers
188 views
+100

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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1answer
71 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 ...
0
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1answer
30 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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9answers
625 views

Learning Model Theory

What books/notes should one read to learn model theory? As I do not have much background in logic it would be ideal if such a reference does not assume much background in logic. Also, as I am ...
2
votes
2answers
78 views

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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1answer
66 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 ...
8
votes
1answer
92 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 ...
3
votes
3answers
97 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). ...
1
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1answer
43 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 ...
2
votes
2answers
105 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 ...
3
votes
1answer
86 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 ...
2
votes
1answer
42 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 ...
1
vote
1answer
50 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 ...
0
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1answer
32 views

Definition of dicrete ordering

What means by "discrete ordering"?
0
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1answer
27 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 ...
3
votes
3answers
258 views

Interpretation of a formula and truth

I just started self-studying Mathematical Logic by Ebbinghaus. I already knew something about formal languages, but nothing about model theory. There is something I don't understand: Exercise 3.3, ...
0
votes
1answer
55 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 ...
2
votes
1answer
31 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 $n,m \in \mathbb{N}$: ...
5
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1answer
398 views

A logic that can distinguish between two structures

it's known that there are non-isomorphic structures that satisfy the same first-order sentences. Likewise it's known (by cardinality arguments) that there are non-isomorphic structures that satisfy ...
0
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0answers
31 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 ...
1
vote
1answer
47 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 ...
0
votes
1answer
46 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 ...
1
vote
2answers
24 views

What exactly is $L$-terms in model theory?

I got confused after seeing the inductive definition of $L$-terms in model theory. So I do get that there are variables and constants, and when function $f$ is applied to the term, the resulting thing ...
1
vote
1answer
35 views

Clarification regarding inner model, standard model, transitive model and Mostowski

After reading books before lectures, here's my thought regarding inner models and so on. Correct me if I am wrong. So there's universe $V$, which we assume to be the true universe. By Gödel's ...
5
votes
2answers
63 views

Removing sets from models of set theory

I have a naive and open-ended question: How can one remove a set from a model of set theory in such a way that the result is again a model of set theory? Directly related: what kinds of sets can ...
0
votes
1answer
46 views

A finite set of wffs has an independent equivalent subset

This seems to be a common exercise among many books (cf. Enderton, p. 28, van Dalen, p. 45, Hinman, p. 51, Chang and Keisler, p. 18), with some minor variations among them. The idea is simple. Say ...
2
votes
1answer
32 views

If $\phi$ is satisfiable and $\mathscr{S}$ is countable, then the set of all models of $\phi$ has the cardinality of the continuum

I have just started reading Chang and Keisler and I'm already stuck in an exercise. Let $\mathscr{S}$ be a countable set of sentence letters (i.e. $\mathscr{S} = \{S_0, S_1, S_2, \dots\}$ or some ...
1
vote
1answer
81 views

Is there a first order theory for equivalences classes?

Question will be a bit naive, so please, be kind. Consider a first order theory, $\Gamma$ . Let $\mathcal{M}$ be the category of models for $\Gamma$. Consider $\sim$ an equivalence relation on ...
3
votes
1answer
51 views

isomorphism between divisible, totally ordered, abelian groups

Let $G$, $H$ be divisible, abelian, linearly ordered groups, whose cardinalities are equal and satisfy $\mu := |G|=|H|>\aleph_{0}$. These are supposed to be (order!) isomorphic. And just about ...
3
votes
1answer
63 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 ...
4
votes
1answer
66 views

Infinite linear order with endpoints which is non-dense

In the process of answering questions about normal models, I had to prove the following: Any normal model of $\chi$ is a non-dense linear order with a least and greatest element. The next question ...
2
votes
1answer
26 views

Extending Henkin's Theorem to Completeness in Marker's Text

In Marker's Model Theory, starting on p. 35, he proves the following: Theorem [Henkin]: If $T$ is finitely satisfiable, then $T$ is satisfiable. He also mentions Theorem [Goedel]: If $T$ is ...
4
votes
1answer
198 views

Construction of *ZFC

In the following paper, page 11 (Appendix), there is a construction of a model of a theory $^*ZFC$ (see the definitions in the paper included) from a model of $ZFC$. I have been trying really hard to ...
2
votes
2answers
57 views

Omitting types and real closed fields

Let $T$ be a theory in a language $L$ such that a model of $T$ defines a real closed field. Can we apply the omitting types theorem in order to show the existence of atomic models of $T$? i.e. does ...
1
vote
1answer
31 views

Propositional S5: is there a consistent set requiring continuously many worlds?

A recent question asked whether in systems of modal propositional logic having the "finite model property" there are consistent sets of sentences that were not satisfied by a finite model. @Carl ...
4
votes
0answers
75 views

Gödel's Completeness Theorem and logical consequence

At the end of a long process of "rumination" on "old" math log textbooks, I've found the "missing link" - from my personal point of view - between some issues I've raised in the previous months : (i) ...
0
votes
1answer
41 views

Propositional modal logic: infinite models required in systems with finite model property?

A system of propositional modal logic has the "finite model property" if any consistent sentence is satisfiable at a model with finitely many possible worlds. Some systems have this property and ...
4
votes
1answer
99 views

Spectrum of elements in a set

Suppose that $X$ is a spectrum. Is $\mathbb{N}\setminus X$ a spectrum? By spectrum, we mean that it is the set containing all natural numbers $n$ s.t. there is a model of $\phi$ with exactly $n$ ...
11
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3answers
370 views

On Pudlak's “Life in an Inconsistent World”

In his Logical Foundation of Mathematics and Computational Complexity (2013), Pavel Pudlak invites the readers to ponder about fictitious people whose natural numbers are nonstandard. His exposition ...
2
votes
1answer
49 views

Countable transitive model of ZFC that is well-founded externally

As I am studying set theory, I came to realize that there exists a countable "well-founded" model of ZFC. But I am curious whether countable models can ever be well-founded externally. What would be ...
5
votes
5answers
329 views

Are there finitely many interesting theorems? [closed]

I'm not a logician, I read, a long time ago, about Gödel etc... A theorem is a provable proposition under a system of axiom. Let's take the usual system of ZFC. Of course, the notion interesting is ...
13
votes
3answers
583 views

Comparing Category Theory and Model Theory (with examples from Group Theory).

The following question eats my brain: The standard definition of a "category" and a list of examples following this definition confuses me. My question is simple: (Q1)If someone write "the category ...