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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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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260 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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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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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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236 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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281 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 ...
2
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2answers
136 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 ...
3
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80 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 $>\omega_{1}$. ...
2
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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 ...
2
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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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1answer
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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46 views

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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1answer
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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1answer
42 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. ...
2
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1answer
90 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 ...
2
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1answer
55 views

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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79 views

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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1answer
108 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
50 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
89 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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3answers
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 ...
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74 views

Lowering the power of infinite model

I need to prove that for every infinite model $\mathfrak A$ of signature $\sigma$ exists model $\mathfrak B$ with attributes: $\mathfrak A \equiv \mathfrak B$. $\parallel \mathfrak B \parallel = ...
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1answer
79 views

Order type of standard models of arithmetic

The standard model of PA has order type $\omega$. By compactness PA has a model of order type $\omega+n$ for any $n$, since every finite subset of the following set of statements is provable: ...
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83 views

Building sequence of axiomatizable classes that…

Any tips how to build sequence of axiomatizable classes $\mathrm{K_0, K_1, ..., K_n, ...}$ that class $\bigcup_{n\in w}\mathrm{K_n}$ is not axiomatizable?
3
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1answer
110 views

Elementary Submodels in Set Theory

I was reading the following summary on elementary submodels: http://boolesrings.org/mpawliuk/2012/01/26/a-practical-guide-to-using-countable-elementary-submodels/ Say $M\prec N$. The link above ...
2
votes
2answers
159 views

For types, is being definable is strictly stronger than being isolated?

Let $A$ be an $L$-structure and let $X\subset A$ and $a\in A$ be a tuple, and consider the type $p=\mbox{tp}_A(a/X)$. Let $T$ be the theory of $A$. Definition 1: $p$ is said to be isolated if for ...
2
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92 views

Some theorems of model theory

Do theorems like "omitting types theorem", "Extended completeness theorem" etc.. hold inside arbitrary countable transitive models of ZFC?
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58 views

Space of countable models of a theory T as a Polish space

Someone told me recently that the space of countable models of a first order theory $T$ form a Polish space. Can some one describe this construction to me?
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81 views

Is every model of $\Gamma$ a model of $Cn(\Gamma)$?

Is every model of $\Gamma$ a model of $Cn(\Gamma)$ ? $Cn(\Gamma)=\{\sigma:\Gamma \models \sigma\}$ This is the set of all sentences logically implied by $\Gamma$ . This could help me to understand ...
3
votes
1answer
78 views

First order logic: intersection is infinite

I am trying to solve my friend's homework assignment, I got stuck at this part: Let $\mathcal{L} = \{P^1, P^2, P^3, \cdots\}$ be language with equality, where $P^i$'s are unary predicates (relation ...
2
votes
1answer
66 views

Possible Turing degrees of countable models of ZFC

Let $M$ be a countable model in a signature $\Sigma$. We assume $\Sigma$ is finite, and (for convenience) has no function or constant symbols. Without loss of generality, we can assume that $M$'s ...
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2answers
104 views

Ultrafilter problem [duplicate]

could you help me with this problem, please? If $U$ is a principal ultrafilter on $I$ such that $\{a\}\in U$. Show that $Ult(\mathfrak{A}_x:x\in I)$ is isomorphic to $\mathfrak{A}_a$ and $[f]=f(a)$ ...
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1answer
131 views

A finite subset of a countable, $\aleph_0$-categorical model invariant under automorphsims is definable? [closed]

sorry to bother you, but I got another question. I appreciate all your comments. Thanks a lot. Let $\mathfrak{A}$ be countable and ${\aleph}_0$-categorical. If $X \subset |\mathfrak{A}|^n$ is ...
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2answers
140 views

Model Theory problem

Good afternoon. I need some help with this little problem. I hope somebody could help me. Thanks a lot Assume that $A\equiv B$. Then there exists a $C$ such that $A\prec C$ and $B\prec C$.
2
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1answer
119 views

Show that if U is a principal ultrafilter then the canonical inmersion is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$

Show that: If $U$ is a principal ultrafilter, then the canonical inmersion $j$ is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$
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1answer
118 views

Show that $\Gamma$ is $\kappa $-categorical for $\kappa>\aleph_{0}$ , but not $\aleph_{0}$-categorical.

Let $\mathcal L=\{c_{i}:i<\omega\}$ be a language in first order logic, and: $\Gamma =\{\forall x(x=x),\forall x\forall y(x=y\rightarrow y=x),\forall x\forall y\forall z(x=y\wedge y=z\rightarrow ...
2
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1answer
129 views

Set of formulas in Model Theory

I'm reading the book Model Theory by Chang and Keisler and there is one thing that always bugs me. Very frequently we have something like $\Sigma(x)$ representing the set of all formulas in a language ...
3
votes
1answer
87 views

Model theoretic answer for having algebraic closure

I am beginner at the model theory and I learn compactness theorem at the class and I saw some application of it and one of them is that "every field has an algebraic closure". How can I prove it with ...
2
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0answers
112 views

Recursive non-standard models?

Any algebraically closed field (ACF) is a model of Modular Arithmetic (MA). MA has the same axioms as first order Peano arithmetic except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. MA ...
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114 views

Compactness Theorem explanation

Compactness Theorem definition: If $T$ is a theory in a first-order language $L$, then $T$ has a model iff every finite subset $S$ of $T$ has a model. A number of questions regarding to this ...
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1answer
42 views

Why do we need sometimes other structures than mentioned in the theorem to prove theorems?

For example when one proves that the elementary theory of finite fields is decidable, one uses pseudo-finite fields which are not in generally finite fields. Why do we need such a larger fields to ...
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1answer
47 views

2-type not-realised in Q

my question is the following: given the additive group of rational numbers, i.e. $Q = \langle {\mathbb Q},+,0\rangle$ and $T$ the theory of $Q$, how can I find (explicitly) a 2-type which is not ...
3
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1answer
72 views

Finitely many countable models implies decidability

Suppose $T$ is decidably axiomatizable first order theory and has no finite model. We shall focus on countable models. If $T$ has just one countable model (up to isomorphism), which means $T$ is ...
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1answer
77 views

Can axioms of the Euclidean space be proven in the Real space?

I'm not a mathematical logic student so my question can be naïve. I'm thinking if we identify $\Bbb{E}^2$ with $\Bbb{R}^2$( as a normed space with the Euclidean norm), Then can we prove all Euclidean ...
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1answer
102 views

A Question Regarding Uncountable Standard Models of ZFC Where CH is False

Let M be an uncountable standard model of ZFC, let $\frak c$ be the cardinality of the continuum, and let (just for the sake of argument) $\mathfrak c=\aleph_2$. If one assumes M has 'all' the ...
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1answer
82 views

Showing a Theory $T$ is Substructure Complete

Let $T$ be a (complete and consistent) theory. Suppose $T$ exhibits the following two properties: (1) model-completeness: if $\mathcal{M} \models T$ and $\mathcal{A} \subseteq \mathcal{M}$ s.t. ...