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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Why is a commutative ring with an infinite number of idempotent elements unstable?

In a book of model theory I found the following statement: A commutative ring with an infinite number of idempotent elements unstable. I haven't manage to prove it yet. As stability in the model ...
5
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1answer
76 views

A question about $R^\infty$-rank

I have been trying to learn some stability theory lately, and I have been reading "Geometric Stability Theory" by Pillay. There is a lemma he states without proving which I am trying to prove, but I ...
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3answers
135 views

Superstructure with sets as atomic/base entities

This question arose while learning nonstandard analysis. The superstructure $V(X)$ of a nonempty set $X$ is defined recursively: $$\begin{align*}V_0(X) &= X \\ V_{i+1}(X) &= V_i(X) \cup ...
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1answer
60 views

Modelchecking on Automata, $\phi$ not SAT and $\phi \models$ False

Given a formula $\phi$ Is $\phi \models FALSE$ equivalent to $\phi$ not SAT? Or does $\phi \models FALSE$ means that $\phi$ is never $TRUE$ and $\phi$ not SAT means, that there existst at least one ...
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2answers
250 views

Consistency of PA: why other proofs?

Completeness theorem affirms that a formal first order system is consistent iff it has a model. The FOL number theory(PA) or First Order Arithmetic has a model, which is the natural numbers structure. ...
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3answers
314 views

Why does creating a model show consistency?

As per the title, why does the ability to generate a model from axioms prove they are consistent?
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1answer
186 views

Characterization of totally categorical theories

I have what I am sure is a trivial question, but I can't seem to answer it for myself. In model theory, there is a theorem of Hrushovski which shows that if T is a totally categorical theory (i.e., T ...
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2answers
165 views

Conservative extension is equiconsistent?

Is it true that a conservative extension of a theory is equiconsistent with it, and, if so, why? WP says: "In mathematical logic, a logical theory T2 is a (proof theoretic) conservative extension of ...
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3answers
364 views

How exactly do logic and mathematics interact and what happens when we change the logic?

[Note: this question turned out to be pretty huge, so if you think it would be better to split it up into smaller questions, please comment. The questions here are quite conceptually intertwined and ...
5
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1answer
211 views

Can locally finite structures really be defined as those structures omitting a certain quantifier-free type?

My question is 2.3.8(b) in Hodges' A Shorter Model Theory, p.43. (It's also in his original Model Theory on p. 47). I'll cite the problem, follow it by definitions of key terms, and then explain why ...
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1answer
145 views

number of linear orders

It is well known that for every infinite cardinal $\kappa$ the number of non-isomorphic total orders of cardinality $\kappa$ is $2^\kappa$. Who first proved this, and in what context? Was it proved ...
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1answer
144 views

Why do $\omega$-models of subsystems of $\mathsf{Z}_2$ satisfy full induction?

Richard Shore, in his 2010 paper in the Bulletin of Symbolic Logic, 'Reverse Mathematics: The Playground of Logic', writes that Obviously, if an $\omega$-model $\mathcal{M}$ (those with $M = ...
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1answer
118 views

Give an example of a homogeneous, but not strongly homogeneous model

To avoid any confusion, the notions of (strong) homogeneity as I understand them are as follows: a model $M$ is said to be homogeneous if for any elementary partial function $f$ from the model into ...
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1answer
116 views

What is the minimal axiomatization of a set of structures?

I wonder what the minimal axiomatization of a set of structures mean? I came across this term from Wikipedia: For a theory $T\in A,$ let $F(T)$ be the set of all structures that satisfy the ...
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1answer
524 views

Non-ZFC set theory and the hyperreals: problem solved?

The reals are the unique complete ordered field. The hyperreals $\mathbb{R}^*$ are not unique in ZFC, and many people seemed to think this was a serious objection to them. Abraham Robinson responded ...
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0answers
317 views

Structuralist slogans

I am afraid to make a bad impression by misusing this forum but I am looking for as-many-as-possible mathematically inspired formulations and references to one (sometimes vague) idea. The idea is ...
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votes
5answers
535 views

Is there a first-order-logic for calculus?

I just finished a course in mathematical logic where the main theme was first-order-logic and little bit of second-order-logic. Now my question is, if we define calculus as the theory of the field of ...
6
votes
1answer
302 views

Topological spaces as model-theoretic structures — definitions?

How do model theorists treat topological spaces as structures, i.e., what are the options for the domain, relations, and operations? I've never done any topology, so I have only the definition to go ...
6
votes
1answer
340 views

Inaccessible cardinals and well-founded models of ZFC

This was a problem in an exam I took last semester, but I never got the chance to ask my professor how to solve it. Here goes: Let $\kappa$ be an inaccessible cardinal. Then $V_\kappa$ is a ...
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1answer
63 views

Trying to construct a model for this algebraic specification

I'm studying for an exam on lambda-calculus and algebraic specifications, and I'm having trouble figuring out this problem. I was wondering if anyone here could help?? The given specification: S: ...
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2answers
172 views

How to prove exact number of congruences over $\mathbb {R}$?

I have to prove that there are exactly 2 congruences over $\mathbb {R}$ seen as a model/structure $\tau = (\varnothing, {+,*}, \varnothing, \operatorname{arity}(+) = \operatorname{arity}(*)=2)$ where ...
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votes
6answers
1k views

What is an efficient nesting of mathematical theorems?

Various mathematical areas of research evolved from a wide and diverse range of questions. Many are physical in nature or come from informatics/computer science, some are procedural or optimization ...
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0answers
251 views

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 ...
3
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1answer
323 views

Model Theory: Homomorphism preserves positive L-formulas

A formula $\varphi$ of a language $L$ is positive iff it can be obtained from atomic formulas by using $\vee, \wedge$. Let $M,N$ be $L$-structures and $f: M \rightarrow N$ be an $L$-homomorphism. How ...
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2answers
240 views

Fraïssé limits and groups

I was recently reading up on Fraïssé limits in Hodges' "A shorter model theory." I was trying to think of some examples and wanted to see if I could take the Fraïssé limit on the category of finite ...
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1answer
232 views

Gluing together mathematical structures, how?

By structure, I mean that which is defined here: http://en.wikipedia.org/wiki/Structure_%28mathematical_logic%29 What I'm looking for is a way of gluing together structures so that each structure ...
4
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1answer
280 views

How many countable models of ZFC are there?

If we were looking at just an arbitrary binary relation on a countable set then I guess we would be looking at infinite graphs and those are uncountable. However, ZFC places an extra structure on our ...
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1answer
101 views

Another question on saturated models of ZFC

Let $M$ be a saturated model of ZFC and let $\kappa$ be any cardinal in $M$. Now let $$\begin{align*} p_\kappa(f) &= \{ “f \text{ 1-1 function”} \wedge \operatorname{dom}(f) \subseteq \omega ...
3
votes
2answers
98 views

$\Sigma_1 \cup \Sigma_2$ has a model

Let $\Sigma_1$ and $\Sigma_2$ be sets of $L$-sentences such that no symbol of $L$ occurs in both $\Sigma_1$ and $\Sigma_2$. Suppose $\Sigma_1$ and $\Sigma_2$ have infinite models. Then $\Sigma_1 \cup ...
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1answer
87 views

Poizat's definition of $p$-equivalent $k$-tuples

Maybe this is too trivial a question to be posted anywhere, but anyway. I am reading Poizat's "A Course in Model Theory". In page 4 he defines the notion of two $k$-tuples, each in the universe of ...
2
votes
1answer
128 views

What is a model of a formal system?

Please give the most illuminating example of a model for a formal system, and a simple example of its use. I also wish an example of an interpretation, and what its useful for.
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votes
2answers
283 views

Primes in nonstandard models of PA

What is known about prime numbers in nonstandard models of PA? Restricted to true natural numbers the sets are identical, but does there always exist nonstandard primes? Can we explicitly define one ...
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votes
2answers
181 views

What does a nonstandard proof of Con(PA) look like?

As in Godel's incompleteness theorem natural numbers encode proofs of theorems. Due to Godel's completeness theorem there is a natural number (in some nonstandard model) that proves $Con(PA)$. What ...
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votes
1answer
193 views

Non-triviality of a semigroup and a semiring

Assume we are given an additive semigroup $M$ which we know it is non-trivial i.e. $M\neq \lbrace 0 \rbrace$. Let $R$ be the semiring obtained from adding a multiplication law to the semigroup. Under ...
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votes
0answers
77 views

Definable measure preserving isomorphisms of $p$-adic semialgebraic sets

Consider a $p$-adic field $K$ (finite extension $\DeclareMathOperator{\bQ}{\mathbb{Q}}$of $\bQ_p$) in Macintyre language $\DeclareMathOperator{\cL}{\mathcal{L}}$ $\cL_{\rm Mac}$. Let $Z$ be a ...
14
votes
2answers
410 views

Comparing countable models of ZFC

Let us consider the class $\cal C$ of countable models of ZFC. For ${\mathfrak A}=(A,{\in}_A)$ and ${\mathfrak B}=(B,{\in}_B)$ in $\cal C$ I say that ${\mathfrak A}<{\mathfrak B}$ iff there is a ...
4
votes
1answer
134 views

Computing square roots and calculus

If one were to verify that $$ \sqrt{2} < 3 $$ would the underlying formalisation require a logic more expressive than first-order? Or, is FOL sufficient since real numbers can be formalised in ...
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vote
2answers
261 views

The structure $(\mathbb{Q}, <)$ is O-minimal

I am trying to prove that the DLO $(\mathbb{Q}, <)$ is an $O$-minimal structure, but I am trying to prove this without using the fact that the complete theory of dense linear orderings without ...
1
vote
1answer
219 views

Structure and Formula encoding for Turing Machine

During my study of Finite Model Theory I found that usually purely relational structure say $\mathcal{M} = \langle A, R_1,\ldots,R_k \rangle$ are encoded as ...
2
votes
1answer
128 views

A problem about complete theory

I'm working on the following problem which is exercise 3.5.1 in Rothmaler's Model Theory book. Show that theory $T$ is complete iff $\phi \vee \psi \in T$ implies $\phi \in T$ or $\psi \in T$ (keep in ...
2
votes
1answer
358 views

Formalising real numbers in set theory

If I understand correctly, the real numbers can be formalised in a first-order, like in ZF. However, such a formalisation is not strong enough to ensure that all models of the reals are isomorphic. ...
0
votes
2answers
132 views

Some questions concerning set-theoretic models of first-order theories

Consider a finitely axiomatized theory $T$ with axioms $\phi_1,...,\phi_n$ over a first-order language with relation symbols $R_1,...,R_k$ of arities $\alpha_1,...,\alpha_k$. Consider the atomic ...
2
votes
1answer
240 views

How can i prove that the theory of random graph has a vaughtian pair?

I'm searching for theories that have a vaughtian pair. I've been given a hint, that $T_{RG}$ has at least one. I have also found many theorems stating in which cases a theory has no vaughtian pair, ...
11
votes
4answers
962 views

How can there be genuine models of set theory?

I know that this a beginner's question asked too many times, but I still didn't get an answer which lets me quit asking: Given that a model/interpretation of a theory (in the Tarskian sense) is a ...
5
votes
1answer
418 views

A logic that can distinguish between two structures

it's known that there are non-isomorphic structures that satisfy the same first-order sentences. Likewise it's known (by cardinality arguments) that there are non-isomorphic structures that satisfy ...
7
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1answer
191 views

A compactness problem for model theory

I'm working on the following problem: Assume that every model of a sentence $\varphi$ satisfies a sentence from $\Sigma$. Show that there is a finite $\Delta \subseteq \Sigma$ such that every model ...
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0answers
184 views

Can the reduced product construction generate boolean-valued models?

In model theory, the reduced product construction contains a collection of structures or models, a set I that indexes the collection, and a filter U on I. Ultraproducts are a special case of reduced ...
6
votes
1answer
250 views

Infinite field of characteristic p elementary equivalent to field with transcendental element over prime subfield

I'm trying to show that if F is an infinite field of characteristic p then it's elementary equivalent to a field F' of char p which contains an element transcendental over its prime subfield (the ...
6
votes
2answers
162 views

Axioms for atomless Boolean algebras

I'm embarrassed to be asking, but: "Write down a set of axioms for the theory of atomless Boolean algebras." This is Exercise 1.14 in Chapter 9 of "Models and Ultraproducts" by Bell and Slomson. I'm ...
3
votes
3answers
268 views

Interpretation of a formula and truth

I just started self-studying Mathematical Logic by Ebbinghaus. I already knew something about formal languages, but nothing about model theory. There is something I don't understand: Exercise 3.3, ...