# Tagged Questions

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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### complete, finitely axiomatizable, theory with 3 countable models

Does it exist a complete, finitely axiomatizable, first-order theory $T$ with exactly 3 countable non-isomorphic models? A few relevant comments: There is a classical example of a complete theory ...
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### Non-axiomatisability and ultraproducts

Let $T$ be a first-order theory over a language $L$, and let $\mathcal{M}$ be a subclass of the class of models of $T$. As I understand it, if there is no theory $\hat{T}$ over $L$ whose class of ...
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### What is the degree of a real closure of an ordered field?

Given a subfield $F$ of $\Bbb R$, let the real-closure of $F$ be the smallest subfield of $\Bbb R$ which is real-closed and extends $F$. Since the theory of real-closed fields is complete, this is in ...
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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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### Definable subset of the additive theory of integers

I strongly suspect that the subset $\{ -1,1 \}$ of $(\mathbb{Z},+)$ is not definable, despite being fixed under all automorphisms of the structure. However, I can't seem to be able find a proof. Does ...
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### Infinite palindromes in number of nonisomorphic posets is independent of $\mathsf{ZF}$

The following is an "exercise" in P. Stanley's book "Enumerative Combinatorics": Let $f(n)$ be the number of nonisomorphic $n$-element posets (...) let $\mathcal{P}$ denote the statement that ...
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### Characterization of the First Order Theory of Ordered Abelian Groups via Quantifier Elimination

In the paper "Elimination of Quantifiers in Algebraic Structures" Macintyre, McKenna and van den Dries, proved that every field (ordered field) whose theory admits quantifier elimination in the ...
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### Using the compactness theorem to show a set of first-order formula is equivalent to a set of quantifier-free formula

I am going through some theorems in Hodges' A shorter model theory'' and I have realized that I do not understand a certain argument regarding compactness. My question has two forms, I am sure that ...
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### Transexponential Functions

Recall that $\exp(1,x) = e^x$ and $\exp(n+1,x) = e^{\exp(n,x)}$. Recall that $f(x)$ is transexponential if $f(x)$ is eventually greater than $\exp(n,x)$ $\forall n \in \mathbb{N}$ I am looking for a ...
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### Elementary submodels in stationary logic.

In the paper "Stationary Logic" by Barwise, Kaufmann and Makkai the authors prove that stationary Logic L(aa) has Löwenheim number $\aleph_1$, i.e. every satisfiable set of sentences has a model of ...
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### Axiomatizability of the multiplication of a ring

The operation of ring multiplication is axiomatizable, if we allow ourselves an additional auxillary addition symbol. Just write down the ring axioms in the signature $\{*,+\}$. But could ...
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### ind-completion and functors which are full with respect to isomorphisms

Let $C$ and $D$ be categories and $F:C\rightarrow D$ a faithful functor which is full with respect to isomorphisms. This means that if $a,b\in C$ and $f:F(a)\rightarrow F(b)$ is an isomorphism in $D$, ...
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### Model theory in terms of type spaces/Lindenbaum algebras

Are there any good references that go into some detail of known 'translations' between properties of the type space of a model and the model theoretic properties of the model? All I seem to find are ...
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### bi-interpretability and automorphism groups

Let $M$ and $N$ be two first order structures, say they are countable and $\aleph_0$-categorical. Then $M$ and $N$ are bi-interpretable if and only if their automorphism groups $Aut(M)$ and $Aut(N)$ ...
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### Completing ordered Fields

How do these two forms of completion behave (in NBG) when fields are authorized to be proper classes? $(i)$: Every ordered field has a real closed, algebraic extension. $(ii$): Every ordered field ...
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### Show undecidability by reducing from Hilbert's $10^{th}$ problem

To show that the existential theory of $\mathbb{Z}$ in the language $\{0, 1; +, \mid , \mid_p\}$ (where $x \mid_p y \Leftrightarrow \exists r \in \mathbb{N} : y=\pm xp^r$) is undecidable we have to ...
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### Morley’s Categoricity Theorem for uncountable languages.

Where can I find an accessible exposition of Shelah’s generalization of Morley’s theorem to uncountable languages? (Please, do not answer “Shelah’s Classification Theory”.)
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### Multiplicative reducts of fields an elementary class?

Consider the multiplicative reducts of fields, that is fields except the addition operation is removed. We are considering the signature {*}, where * is an operator of arity 2. Is that class an ...
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### An affine group behaving like a field

This question is about an example of interpreting a field in an affine group, from Section 1.3 of Marker's Model theory: An introduction. Let $F$ be an infinite field and $G$ be the group of ...
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### Countable transitive $T \vDash ZFC-P$ with $\approx$ not absolute: why do we need $H(\aleph_3)$?

I am trying to solve Exercise IV.3.31 from Kunen's Foundations of Mathematics. I think I have a solution but I am confused by one of the hints. By request, here is the text of the exercise. ...
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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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### Defining “structured sets”

In his Notes on Set Theory (p. 44) Moschovakis defines: A structured set is a pair $U = (A,S)$ where $A$ is a set, the space of $U$, and $S$ is an arbitrary object, the frame of $U$. But even ...
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### Crankery: Is there a perfect inner model of ZFC?

In the book "Aspects of Vagueness", the article "The alternative set theory and its approach to Cantor's set theory" by A. Sochor proposes the following definition: We will say that a set universe ...
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### Formalizing model-theoretical large cardinals in a formal system for ZFC

I work with the Metamath formal system, which is expressive enough to define ZFC and prove some nontrivial stuff, but my current goal is to define large cardinals, and I'm hitting a wall somewhere ...
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### What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces?

According to the answer by François G. Dorais, we know that a logic $\mathfrak L$ is compact iff its Stone space of the Lindenbaum–Tarski algebra of the empty theory (w.r.t the deductive system) is ...
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### Some intuition behind o-minimal systems.

I am following the definitions of an o-minimal system as defined in "Tame Topology and O-minimal Structures" by Lou van den Dries. It is immediate from the definition that the graph of $\sin(x)$ is ...
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### Cardinality of Galois groups

We know that no Galois group of a Galois extension is countable. The question is: which cardinalities are possible for a Galois group? (or also: for profinite groups?) I suspect that the theory of ...
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### Defining the Squaring Function in $(\mathbb{Z}; 0, 1, +, -, |)$?

I'm trying to show that the map $x\mapsto x^2$ is 0-definable in the structure $(\mathbb{Z}; 0, 1, +, -, |)$ (group of integers with the divisibility relation). But I'm not sure how to proceed. Any ...
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### Positive existential theory of an extension of the ring

When we know that the positive existential theory of a ring $R[x]$ in a language $L$ is undecidable, does it follow that the positive existential theory of $R[x,y]$ in the same language $L$ is also ...
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### model theory & algebraically closed field

recently I'm taking mathematical logic course, and the class covers some basic model theoretical ideas. Since I have not taken any abstract algebra course, It is so hard to understand what is going on....
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### Principia Mathematica Part VI “Quantity” vs Part IV “Relation Arithmetic”

In "My Philosophical Development", of Principia Mathematica Part IV "Relation Arithmetic", Bertrand Russell laments: "I think relation-arithmetic important, not only as an interesting ...
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### Shepherdson's model for Open Induction

In the paper "A Non-Standard Model for a Free Variable Fragment of Number Theory", Shepherdson constructs a recursive model for a fragment of arithmetic known as "Open Induction". I would like to ...
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### Model-theoretic characterization of local modal correspondence

I've been reading van Benthem's dissertation (available on ILLC's website) on modal correspondence theory. In Section I.3, he develops a model-theoretic characterization of modal formulas having ...
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### Application of the reflection theorem in ZFC

In a proof that a certain theory $T$ in conservative over $\textsf{ZFC}$, the author makes the following step: (Here $\Delta$ is a finite set of formulas such that $\textsf{ZFC}\vdash\Delta$.) ...
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### References for Introductory Model Theory focusing on applications other than algebra

I would appreciate suggestions for references (books, lecture notes, articles etc...) on Model Theory (at an introductory level) that don't focus mostly on algebra when giving examples and/or applying ...
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### Consequence in $\mathcal{L}_{\infty\lambda}$

Consider the infinitary first-order language $\mathcal{L}_{\infty\lambda^+}$ whose non-logical vocabulary consists of $\lambda \geq \omega$ individual constants and countably many predicate constants ...
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### Ordering of $\mathbb{R}$ not quantifier-free definable in $L_{R}$

I'm reading David Marker's book "Model Theory: An Introduction" and I'm trying to solve Exercise 3.4.24 which is stated as follows: Let $x$ and $y$ be algebraically independent over $\mathbb{R}$. a) ...
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### Uncountably Categorical Theories and Embeddings

Suppose that $T$ is uncountably categorical. By the Baldwin-Lachlan Theorem, we note that $I(T,\aleph_0)=1$ or $\aleph_0$. Suppose that $I(T,\aleph_0)=\aleph_0$. Is it always the case that we get ...
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### Prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$ where $\{I_j \ | \ j \in J\}$ is a partition of I.

My problem is following: prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$, where $\langle \mathbf{A}_i \ | \ i \in I \rangle$ is an indexed set of similar ...
Let $T$ be a complete first order theory. Suppose that $M\models T$. Then, $M$ is said to be a Jónsson Model of $T$ if for all $N$, such that $N\prec M$ and $N\models T$, we have $|N|<|M|$ (Note ...
Hello I am having difficulty with this question, I am not even sure what strategy one would go about proving something like this: Suppose $L$ is a language which includes an infinite list \$c_1,c_2,...