# 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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### About a sentence in logic theory that I don't understand.

Can somebody explain to me the following terms in logic? I have to read a paper in combinatorics that says this, but I don't understand anything in this sentence, where the author speaks about logic. ...
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### Is there a Second-Order Axiomatization of ZF(C) which is categorical?

A theory $T$ is called categorical if it only has one model upto isomorphism. (Note: this has nothing to do with category theory.) The Lowenheim-Skolem theorem states that no first-order theory with ...
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### Models of real numbers combined with Peano axioms

Suppose you take the axioms for a Dedekind-complete ordered field and weaken the Dedekind-completeness axiom to the corresponding weaker first-order axiom schema (e.g. replace the left and right sets ...
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### Neither Even Nor Odd Natural Numbers

I confused myself and the OP when I tried to answer a recent question. Modular arithmetic (MA) has the same axioms as first order PA except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. ...
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### Elementary equivalence of polynomial rings

In his notes on the model theory of valued fields, Lou van den Dries mentions in bypassing that the polynomial ring over the complex numbers $\mathbb{C}$ is not elementarily equivalent to the ...
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### An exercise in stability theory

This is taken from Pillay's highly minimalistic book on stability theory. Let $T$ be stable, $\mathcal{M} \prec \mathcal{N}$ models, and $a$ a tuple in the big model such that its type over $N$ ...
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### What are some ways of showing that a structure is not minimal?

The question is really in the title. My background in model theory is very limited. Basically nothing past the definition of minimal structures and minimal subsets. I am interested in some ...
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### Elementry question on elementarily equivalence

Source: SHAWN HEDMAN Definition:Let M and N be V-structures. If M and N models the same V-sentences, then M and N are said to be elementarily equivalent, denoted $M \equiv N.$ Example: the ...
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### Can there be an abelian group $G$ where $\bigcap_{n\in\mathbb N} nG$ is not divisible?

I heard a talk yesterday on MacIntyre's theorems, which involved a decomposition of ($\omega$-stable) groups into a divisible part and a bounded exponent part. Apparently there is a result of ...
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### Theories with countably many countable models

Having another question in mind (which is not yet fully worked out, but will come soon) I'd like to gather some examples of (interesting) theories with countably many countable models ...
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### Number of isomorphism classes of countable models of a theory

Whether there are countably or uncountably many isomorphism classes of countable models of a given theory depends on the theory: if the theory is strong enough, there will be only countably many ...
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### What is a Proper Signature for a Vector Space?

The definition of a signature I'm working with is a quadruple $\sigma = (C,F,R,\sigma')$ with $C$ serving as a set of constant symbols, $F$ serving as a set of function symbols, and $R$ serving as a ...
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### Why doesn't the definition of (model-theoretic) conservative extension need strengthening?

In the wikipedia page dedicated to conservative extensions, we find the following sentence: $T_2$ is a model-theoretic conservative extension of $T_1$ if every model of $T_1$ can be expanded to ...
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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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### Question on Model completeness in FOL

Claim: Suppose T is model complete and has a model embeddable in every model of T. Show T is complete. This is in Sacks' book Saturated Model theory, problem 8.4. Is the following proof correct? ...
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### modeltheory, union

There's a row of models, namely $M_0 \subseteq M_1 \subseteq M_2 \subseteq \dotsb$ and there's a theory $T$ such: $\phi \in T \rightarrow \phi = \forall x\exists y\,\psi$ with $\psi$ quantor-free. ...
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### Can a model of set theory think it is well-founded and in fact not be?

ZF's axiom of regularity implies that no infinite descending sequence of sets $x_1 \ni x_2 \ni x_3 \ni \cdots$ exists. Precisely this theorem asserts the non-existence of a map from $\mathbb{N}$ to ...
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### Renaming the elements of a mathematical structure

One of the most basic insights about mathematical structures is that we can rename their elements without fundamentally changing the structure. Question. How do we actually formalize this observation ...
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### Logic: some basic plane geometry

Suppose you've got the language of some basic plane geometry, i.e. two 1-place relation symbols $P$ and $L$ for point and line and one 2-place relation symbol $I$ for point $x$ lies on line $y$. Now, ...
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### Logic: $\text{Mod } \Sigma$, the class of all models of $\Sigma$ and $\text{Th Mod }\Sigma$, how do these relate?

While reviewing a question I had asked earlier here: Proving and understanding the Fixed point lemma (Diagonal Lemma) in Logic - used in proof of Godel's incompleteness theorem I have the ...
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### Logic: cardinality of the set of formulas

How can you proof that $||L||=|L|$ if $L$ is infinite (where $||L||$ stands for the cardinality of the set of all $L$-formulas and $|L|$ the number of all constants, function and relation symbols)? ...
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### What's the idea behind the proof of saturation of internal sets via ultrapower construction?

I'm trying to understand the proof of saturation of internal sets via ultrapower construction on Robert Goldblatt's Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. Though, it's ...
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### Can I express (only) the syntactical formulation of a set theory within another.

Motivated by this question on the SE philosophy board "Is there a one true set theory", I want to ask for a clarification: I've seen e.g. ZF being expressed in Grothendieck set theory. If I say ...
### Every connected $\omega$-stable group has a zero element?
Let $G$ be a connected $\omega$-stable group and $p$ its unique generic. Let $a$ be a realization of $p$, $G\prec G_1$ an elementary extension containing $a$ and $q$ the non forking extension of $p$ ...
Let $E$ be a Banach space and $(E)_{\mathcal U}$ be an ultrapower for some ultrafilter $\mathcal U$ on an index set $I$. It is remarked in a paper that $(E')_{\mathcal U}$ can be naturally embedded ...