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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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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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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45 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 ...
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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 ...
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103 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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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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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 ...
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89 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 ...
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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 ...
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92 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 ...
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77 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 ...
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74 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 $\langle Q,+,0\rangle$.
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86 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 ...
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1answer
42 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 ...
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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) ...
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81 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 ...
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1answer
88 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 ...
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390 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 ...
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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 ...
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230 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 ...
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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 ...
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2answers
159 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 ...
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39 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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111 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, ...
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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) ...
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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 ...
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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 ...
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1answer
100 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 ...
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1answer
66 views

Can integers be defined in the first-order theory of the rationals?

Can integers be defined in the first-order theory of the rationals with addition, multiplication, and order?
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34 views

Are all first-order truths of real arithmetic also true of the algebraic reals?

Consider sentences in first-order logic which are true of the structure $(\mathbb R, +, \cdot, <, 0, 1)$, where the symbols have their usual meaning. Is every such sentence also true of $(\mathbb ...
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76 views

Theory of Fields $\omega$-Inconsistent?

A theory is $\omega$-inconsistent if there is a predicate $P(n)$ that is true for every standard natural number yet not true for all numbers. Consider the theory of finite fields and let $P(x) = (Sx ...
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64 views

Sub-models which are elementary equivalent, but not elementary submodels.

I have been trying to solve the following problem from Marker's Model Theory. Find a pair of models $\mathcal{M}$ and $\mathcal{N}$, and a subset $A\subseteq \mathcal{M}$ so that 1) ...
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95 views

Isomorphism type of strongly minimal sets in an uncountably categorical theory

Let $T$ be an uncountably categorical theory in a countable language. If $\mathcal{M}$ is a model of $T$ then there is a strongly minimal definable subset $D$ of $\mathcal{M}$ such that the dimension ...
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1answer
126 views

A model of geometry with the negation of Pasch’s axiom?

How would a geometry with all the usual axioms of Euclidean geometry, except that instead of Pasch’s axiom, we take it negation, look like?
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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 ...
3
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1answer
73 views

Models of $T_{\forall\exists}$ embed in a existentially closed extension model of $T$.

In order to prove (the non trivial part of) Chang-Los-Suszko's theorem [1], I'm struggling with the following lemma : Lemma. Let $T$ be a $\mathcal L$-theory and $T_{\forall\exists}$ the set of ...
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1answer
75 views

Exercise 17.6 of Sacks' Saturated Model Theory

I'd like to know whether my proof is correct. Exercise goes as follows. 17.6. Let $T$ be a model completion of some $\forall$-theory. Show there exists $T^* = T$ s.t. every member of $T^*$ is of the ...
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1answer
235 views

Recommendation on a rigorous and deep introductory logic textbook

In this post, I don't mean any word by its somewhat "mathematical or logical" meaning but just "literally". It's been three years since I started "formal" mathematics, and now I'm familiar with set ...
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195 views

A truth definition, wrong, but where

Consider the theory ZFC of language with connectives $\neg$, $\rightarrow$, predicative symbol $\in$ and quantifier $\forall$. Assume that we are working in ZFC. Let $\mathcal{V}$ be the class of ...
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1answer
102 views

Exercise 16.7 from Sacks' Saturated Model Theory

Question / exercise goes as follows: $M'$ is said to be finitely generated if there exists a finite $|X|\subset |M'|$ such that $M'$ is the least substructure of $M'$ whose universe $|M'|$ contains ...
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48 views

How much does a theory tell about categorical properties of its models.

Let take a first order theory $T$ in a language $\mathcal L$. One can form the category $\mathrm{Mod}(T)$ whose objects are the models of $T$ (those $\mathcal L$-structures $\mathfrak M$ such that ...
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Semantic Proof of Tarski's Undefinability of Arithmetic Truth

A few years ago I took a logic course and I've since lost my notes. I seem to remember a very semantic proof of Tarski's theorem on the undefinability of arithmetic truth, one that didn't use the ...
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3answers
72 views

Satisfiability Proof Question

Exercise: Prove that $\Gamma\models A$ iff $\Gamma\cup\{\neg A\}$ is not satisfiable. Proof: We must prove two clauses: $\Gamma\models A\Rightarrow \Gamma\cup\{\neg A\}$ is not satisfiable ...
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Question from Marker's book T3 has three models up to isomorphism

This question is from Marker's book. Let $ \mathcal L_3 = \left\{ {< ,c_0,c_1, \dots}\right\} $ where $c_0,c_1, \dots$ are constants symbols. Let $T_3$ be the theory of dense linear orders ...
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66 views

Algebraic types are isolated

Let $\mathcal M$ be a $\mathcal L$-structure, and $A\subseteq M$. Let $p(x)\in S_1(A)$ be a complete 1-type of the theory $T_A := \mathrm{Th}(\mathcal M_A)$. We say that $p$ is algebraic if there ...
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Saturation, (Complete) Ordered Fields and Model-Theoretic Methods in relation to Real & Non-Standard Analysis

I am trying to understand the following three questions: One and Two and Three. I'm under the impression that they're interrelated, though maybe not directly. What do I need to read to back-fill to ...
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1answer
45 views

Let $T$ be a theory of Abelian groups where every element has order 2 find complete theory include T

This question is from Marker's book . Let $T$ be a theory of Abelian groups where every element has order 2 . Show that it is not complete . Find $T' \supset T $ a complete theory with the same ...
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Show $\forall x \exists y F(x,y)$ does not imply $\exists y \forall x F(x,y)$

Show: $\exists y \forall x R(x,y) \rightarrow \forall x \exists y R(x,y)$ $\forall x \exists y F(x,y)$ does not imply $\exists y \forall x F(x,y)$ How do proofs of this nature usually work? When I ...
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143 views

Tennenbaum's theorem without overspill

While trying to clean up Wikipedia's proof sketch for Tennenbaum's theorem (there is no computable non-standard model of Peano Arithmetic), the following strategy occurred to me. Since it seems to be ...