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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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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101 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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24 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) ...
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48 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 ...
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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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93 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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95 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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63 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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74 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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53 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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88 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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108 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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127 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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64 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
67 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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175 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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91 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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47 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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81 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
66 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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57 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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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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134 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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1answer
134 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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0answers
37 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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1answer
57 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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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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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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1answer
101 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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100 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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2answers
257 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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1answer
136 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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43 views

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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234 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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2answers
280 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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1answer
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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2answers
131 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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1answer
78 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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1answer
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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2answers
62 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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1answer
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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1answer
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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$(M, <) \equiv (\mathbb{Z}, <)$ 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. ...