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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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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75 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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59 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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89 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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114 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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138 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 ...
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68 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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73 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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208 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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97 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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90 views

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
70 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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55 views

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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58 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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82 views

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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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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147 views

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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139 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 ...
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39 views

Books/papers on model theory in non-monotonic logic

I am working on a project whose object language is in non-monotonic logic. Since the project involves reasoning about the models, I am thinking of translating a non-monotonic problem into a ...
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58 views

Problem concerning formally real fields

I'm trying to reconcile a fact I am reading in David Marker's Model Theory text. He claims on page 326 that $\mathbb{F}=\mathbb{Q}(\sqrt{2}, \sqrt{-2})$ is a formally real field. This seems like it ...
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48 views

Categorical description of permutation-invariance of models

One of my projects (which I should really leave to professionals, but I'm trying anyway) is trying to find a categorical description of the permutation-invariance of models of NF, if there is such a ...
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160 views

Functoriality of the correspondence between oligomorphic actions and $\aleph_0$-categorical theories

If a group $G$ acts on a set $X$, then the action is said to be oligomorphic if the number of orbits of $X^n$ under the action is finite for each $n$. There is a classic theorem in model theory that ...
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104 views

Testcases for Baldwin Lachlan

Let $T$ be a complete theory in a countable language with infinite models. By a theorem of Baldwin-Lachlan, $T$ is uncountably categorical if and only if it is $\omega$-stable and has no Vaughtian ...
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103 views

Isomorphism of finite models

Let $\mathfrak A$ and $\mathfrak B$ are models of finite signature $\sigma$. Prove that $\mathfrak A$ and $\mathfrak B$ are isomorphic, if $\mathfrak A \equiv \mathfrak B$ and $\mathfrak A$ is ...
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275 views

Number of Non-isomorphic models of Set Theory

Assume that the meta theory allows for model theoretic techniques and handling infinite sets etc (The meta theory itself is, informally, "strong as ZFC"). Also assume that I'm studying ZFC inside this ...
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138 views

How do we know that certain concrete nonstandard models of the natural numbers satisfy the Peano axioms?

It is easy to come up with objects that do not satisfy the Peano axioms. For example, let $\Bbb{S} = \Bbb N \cup \{Z\}$, and $SZ = S0$. Then this clearly violates the axiom that says that $Sa=Sb\to ...
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If A is a model of (B is a model of C), then can we say that A is a model of C?

The model relationship "$\models$" is first order definable, so one could have a sentence S which said that a structure B is a model of a sentence C, and S could in its turn be satisfied by a model A. ...
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242 views

How best to formalize propositions suffering from “size issues”?

Suppose we want to formalize a proposition (say, from category theory or model theory) that has "size" issues. For concreteness, lets take the following statement as a fairly typical example. ...
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285 views

Can A Decidable Theory Have Nonrecursive Models?

Tennenbaum's theorem proves neither addition nor multiplication can be recursive in any countable non-standard model of arithmetic. Tennenbaum's proof applies to theories much weaker than PA. ...
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53 views

Curve in $(\mathbb{R},<)$ going to infinity

My question is the following: Given the structure $(\mathbb{R},<)$ and $t \in \mathbb{R}$, can I have a definable function $f$ over a finite set of parameters, with domain $(-\infty, t)$ and with ...
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139 views

Pseudo-finite field vs Nonstandard finite field

Let $\mathbb{N}^*$ be a countable non-standard model of Peano arithmetic (PA) and let $\mathbb{Z}^*$ be the integers extended from $\mathbb{N}^*$. A non-standard finite field would be a ring ...
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84 views

Does model theory extend to partial functions?

I have been reading a bit about effect algebras and d-posets recently, sets $M$ on which you have a single partial binary operation (partial here meaning partially defined, i.e. the domain of this ...
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40 views

Examples of $\omega_{1}<\alpha<\omega_{2}$ definable

Can you think of any examples of definable ordinals between $\omega_{1}<\alpha<\omega_{2}$? I am trying to show that countable $M\prec (H(\aleph_{2},\in)$ contains ordinals $>\omega_{1}$. ...
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66 views

every element of $V_{\omega}$ is definable

My attempt by $\in$-induction. I am trying find formula that will work: $N=(V_{\omega},\in)\models rank(\varnothing) =0<\omega$ Assume,given $x\in V_\omega$ that $\forall y\in x$ are definable too ...
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57 views

Determining if a theory in first-order logic is decidable

We have a theory in first-order logic which we know that is uncountably categorical, complete but not finitely axiomatisable. We also want to know if it is decidable. But I don't know the procedure ...
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137 views

Questions on a sentence in ZFC asserting that ZFC has a model

Question 1. Let $\operatorname{con}(\mathsf{ZFC})$ be a sentence in $\mathsf{ZFC}$ asserting that $\mathsf{ZFC}$ has a model. Let $S$ be the theory $\mathsf{ZFC}+\operatorname{con}(\mathsf{ZFC})$. ...
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Prove that the set of $\mathcal{L}$-terms has size $\max\{\aleph_0, \#\text{ of constant sym plus the $\#$ function sym}\}$.

Let $\mathcal{L}$ be a first-order language. Prove that the set of $\mathcal{L}$-terms has size $\max\{\aleph_0,\text{ the number of constant symbols plus the number of function symbols}\}$. I know ...
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44 views

Prove that $\Phi_{eq}$ has continuum many closed complete extensions.

Full question: Let $\mathcal{L} = \{E\}$ where $E$ is a binary relation and let $\Phi_{eq}$ be the axioms for an equivalence relation. Prove that $\Phi_{eq}$ has continuum many closed complete ...
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89 views

Embedding models of ZF into another model

I had some ideas regarding models of ZF. My ideas (phrased as questions) are: Given two models of ZF, what are the condition for a model containing both models (in the sense of embedding) to exist? ...
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47 views

$(M, <) \equiv (\mathbb{Z}, <)$ and $(\mathbb{Q}, <)$ embeds into $(M, <)$

This is homework for a class I didn't take, but which is a prerequisite for a course I will take. In particular I supposed it's something other people may see as a homework problem in the future. ...
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91 views

Power set axiom

I want to write it in the following language $L=\{\in,=\}\cup \{u_{0},u_{1},\ldots\}$. Here is an attempt $\forall x~\exists y~\forall z~\in x~[z=p\wedge p\subset y]$. I also want to show whether it ...
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Elementary submodels of $(V_{\omega},\in)$ are equal to it

I read that all the ESMs of $(V_{\omega},\in)$ will be equal to it . But what if the ESM's universe is a finite set?
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Is there a reason to consider formulas with multiple quantifiers on the same variable?

I'm very green at mathematical logic, so I apologize for what may be a stupid question. As I understand it, the definition of a first-order formula allows for monstrosities like $$\exists x \neg ...
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112 views

Do metatheoretic results carry between mutually interpretable theories?

If two theories A and B are mutually interpretable, in the sense of there existing a translation procedure from A to B and from B to A, does it follow that whatever metatheoretic results (e.g., ...
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1answer
54 views

Prove the elementary equivalence of the two models

There are two models $\mathfrak A$ and $\mathfrak B$ in class $K$. $\mathfrak A = <P(\omega), \subseteq>$ $\mathfrak B = <P(\omega), \supseteq>$ Is the $Th(K)$ of a full theory of ...
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1answer
90 views

Why is class of one-dimensional vector spaces not axiomatizable?

Prove, class of all one-dimensional vector spaces over R isn't axiomatizable in the signature σ=< +, α, 0>, α* is single function vector multiplication by a scalar $\alpha$ of R. There are ...
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46 views

truth of a sentence to the linearly ordered set

Let Φ - sentence of signature σ = <≤> such that for any infinite linearly ordered set A satisfies A ⊨ F. Prove that there exists n ∈ N such that for every linearly ordered set B power greater than ...