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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259 views

how to show that a group is elementarily equivalent to the additive group of integers

Is there any fairly easy way of showing a group is elementarily equivalent to the additive group of the integers? I've found a simple characterization here: A ‘natural’ theory without a prime model, ...
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1answer
93 views

Proof through consistency

Take first-order Peano Arithmetic PA. We know that Gentzen proved PA consistent. Now, if one sets for example $\varphi$ to represent Fermat's theorem in FO, would proving PA+$\varphi$ consistent be ...
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3answers
186 views

Non-Archimedean non-standard models for R

Let $\langle R,0,1,+,\cdot,<\rangle$ be the standard model for R, and let S be a countable model of R (satisfying all true first-order statements in R). Is it true that the set 1,1+1,1+1+1,… is ...
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2answers
204 views

Is this incompleteness result easier to get than incompleteness of PA?

Gödel's theorem for Peano Arithmetic shows that (under consistency hypothesis on PA) there is a statement which cannot be proved or disproved within PA that is true under the standard model ...
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4answers
246 views

Theories and models

I apologize if my question is not well formed. The reason for it is that I don't understand the concepts enough to be able to ask a completely meaningful question. In the classes we said that a ...
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3answers
2k views

Is Foundational Research a Dead Field?

I'm a second year mathematics major at a pretty good school. Ever since I became a math major I have been most interested in set theory and logic, which I guess can be lumped into the category of ...
3
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1answer
125 views

Automorphisms of elementary extensions

I think this is probably a very simple question, but I've been puzzling over it for a while and can't seem to get anywhere. Suppose $M$ is a structure, $\alpha$ is an automorphism of $M$, and $N$ is ...
2
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2answers
306 views

Lowenheim-Skolem theorem and first-order model

In Wikipedia, it says that a nonstandard model of natural numbers is not first-order. But, from the Lowenheim-Skolem theorem, I don't see anything that points to this conclusion. Can anyone show me ...
2
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1answer
285 views

Mathematical structures and signature

From Wikipedia: In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach (or relate) to the set, making it easier ...
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1answer
695 views

Question about the proof of consistency iff satisfiability of a theory

In pretty much any Model Theory or Logic textbook you will find the following claim, where $T$ is a theory (a set of $\mathsf{L}$-sentences), $T$ is consistent if and only if $T$ is satisfiable. ...
2
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1answer
227 views

Independence results in first-order PA and second-order PA

There are statements $\varphi$ that are independent of first-order Peano Axioms. Are these statements also independent of second-order Peano Axioms? I'm reading Wikipedia articles around ...
2
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0answers
51 views

Nice model theoretic properties of a theory adding a new predicate?

I would like to know what nice model theoretic properties (for example simplicity, NIP, stability, etc) can be preserved when we add a new predicate to the language. Explicitly, if $T$ is an L-theory ...
5
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1answer
344 views

Does the existence of a $\mathbb{Q}$-basis for $\mathbb{R}$ imply that choice holds up to $\frak c$?

The axiom of choice is, for ZF, equivalent to the statement that every vector space has a basis. The implication of AoC by the existence of a basis for any vector space is shown in this paper. The ...
2
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1answer
218 views

What does variety language mean?

In Zilber's paper entitled 'a theory of generic function with derivations', he works in the variety language for fields? What does this mean? If I want to define the variety language what can I say?
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1answer
176 views

I want examples of definable groups in algebraically closed fields?

I can not think of any example of a definable group in algebraically closed fields?
5
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2answers
255 views

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
votes
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 ...
5
votes
3answers
136 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
251 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. ...
11
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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?
3
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1answer
188 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
167 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 ...
5
votes
3answers
366 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
votes
1answer
216 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 ...
4
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1answer
146 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 ...
3
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1answer
145 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
123 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 ...
3
votes
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 ...
14
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1answer
531 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 ...
5
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0answers
320 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 ...
4
votes
5answers
538 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
311 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
342 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
65 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: ...
1
vote
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 ...
30
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 ...
6
votes
0answers
252 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
votes
1answer
331 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 ...
8
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2answers
242 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 ...
1
vote
1answer
234 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
votes
1answer
281 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 ...
1
vote
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 ...
1
vote
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
129 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.
5
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2answers
287 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 ...
4
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 ...
0
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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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 ...