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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146 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
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2answers
81 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
124 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 ...
11
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2answers
234 views

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
84 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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1answer
36 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 ...
2
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2answers
121 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 ...
8
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1answer
138 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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1answer
90 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 ...
1
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1answer
57 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
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1answer
60 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
137 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
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1answer
126 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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1answer
60 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 ...
4
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3answers
115 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). ...
0
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1answer
31 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 ...
1
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1answer
35 views

Definition of dicrete ordering

What means by "discrete ordering"?
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1answer
60 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
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1answer
41 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 $n,m \in \mathbb{N}$: ...
0
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0answers
54 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
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1answer
80 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
81 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 ...
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2answers
36 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 ...
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1answer
43 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
70 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
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1answer
100 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 ...
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1answer
46 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 ...
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1answer
92 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
78 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
3answers
109 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
87 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
73 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 ...
2
votes
1answer
73 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
80 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
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1answer
41 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 ...
5
votes
0answers
140 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
68 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 ...
2
votes
1answer
83 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 ...
12
votes
3answers
387 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 ...
1
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1answer
56 views

Possible independence of a generic sentence?

Let the generic sentence $P :$ "$\exists z \in Z \text{ such that }p(z) \text{ is true }$". In addition $Z$ is recursively enumerable, and for a given $z_0$ in $Z$, "$p(z_0)$ or $ \lnot p(z_0)$" is ...
4
votes
1answer
227 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 ...
0
votes
0answers
55 views

Non-Standard Arithmetic - order

Recently I try to figure out some facts about one specific way of "constructing" a non-standard model for (peano) arithmetic. I guess there are answer to my question already out there, but somehow I ...
3
votes
2answers
152 views

I Need Help Understanding Quantifier Elimination

I am struggling to understand Quantifier Elimination as it is treated in Hodges' "A Shorter Model Theory". The relevant definitions are : Definition: Take $K$ to be a class of $L$-structures, for ...
1
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1answer
38 views

Questions About Elementary Extensions in Model Theory.

If we have a model $A$ of a first-order signature $L$ with $B$ an elementary extension of $A$, then if we extend $L$ to $L(\bar{c})$ by adding some constant symbols $\bar{c}$ not in $L$, is $(B, ...
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votes
1answer
106 views

Every Countable Model of PA is Recursive?

I am interested in any obvious flaws in the following argument. Assume we have a countable model of Peano arithmetic in a meta-theory like ZFC. Assume this model has a set of ordered triplets, ...
0
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1answer
29 views

Reformulation of Theories

Philosophical questions (or even just a matter of taste) regarding some mathematical constructions can give rise to reformulations of whole theories, for example, we can develop (Non-standard) ...
0
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1answer
49 views

Not Skolem's Paradox - Part 3

This is a follow up to a previous question: Not Skolem's Paradox - Part 2. Assume we have a countable, non-standard model of Peano Arithmetic in ZFC. This ZFC model must include a set of ordered ...
2
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0answers
51 views

The composition of functors and algebraic structures

An algebraic structure can be viewed as a finite-product-preserving functor $X : \mathbf{L} \rightarrow \mathbf{C},$ where $\mathbf{L}$ is a Lawvere theory and $\mathbf{C}$ is a category. Thus given ...
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1answer
94 views

Why is the Ehrenfeucht theory complete?

I am looking at the theory T of Dense linear orders without endpoints, extended with the set $\{c_i<c_j|i\in\omega\}$ and am asked to prove that this theory is complete. I know that it has three ...
2
votes
1answer
98 views

Showing that a theory is complete.

This is a homework question so please don't give the full answer, just an approach will do. Question: Let $T=D_u\cup\{c_i<c_{i+1}\mid i\in\mathbb N\}$ where $D_u$ is the theory of dense linear ...