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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Is there a 0-1 law for the theory of groups?

For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } ...
53
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8answers
5k views

How is the Gödel's Completeness Theorem not a tautology?

As a physicist trying to understand the foundations of modern mathematics (in particular Model Theory) $-$ I have a hard time coping with the border between syntax and semantics. I believe a lot would ...
32
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6answers
2k 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 ...
31
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6answers
2k views

Do the axioms of set theory actually define the notion of a set?

In Henning Makholm's answer to the question, When does the set enter set theory?, he states: In axiomatic set theory, the axioms themselves are the definition of the notion of a set: A set is ...
29
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2answers
604 views

Is $ \pi $ definable in $(\Bbb R,0,1,+,×, <,\exp) $?

Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields together with the unary function symbol $ \exp $ such that in the standard ...
27
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4answers
1k views

Is $\mathbb{N}$ impossible to pin down?

I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical. In ZFC, ...
25
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3answers
1k views

Murder at Hilbert's Hotel!

I'm sorry if this is a duplicate in any way. I doubt it's an original question. Due to my ignorance, it's difficult for me to search for appropriate things. Motivation. This question is inspired by ...
23
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1answer
803 views

Does there exist any uncountable group , every proper subgroup of which is countable?

Does there exist an uncountable group , every proper subgroup of which is countable ?
22
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1answer
474 views

FO-definability of the integers in (Q, +, <)

With $Q$ the set of rational numbers, I'm wondering: Is the predicate "Int($x$) $\equiv$ $x$ is an integer" first-order definable in $(Q, +, <)$ where there is one additional constant symbol ...
22
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1answer
599 views

Model existence for infinitary logics

One of the problems of infinitary logic is that it is possible for compactness to fail in a spectacular way: for example, one can concoct an inconsistent set of axioms whose proper subsets are all ...
20
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1answer
2k views

Non-standard models of arithmetic for Dummies

Why is (1) a copy of $\mathbb{N}$ "followed by" a copy of $\mathbb{Z}$ not a (non-standard) model of arithmetic, neither (2) a copy of $\mathbb{N}$ followed by an infinite sequence of copies of ...
19
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2answers
1k views

Are there number systems corresponding to higher cardinalities than the real numbers?

As most of you know, the set $\omega$ with cardinality $\aleph_0$ corresponds to what we normally know as the natural numbers $\mathbb{N}$, and the set $\mathcal{P}(\omega)$ with cardinality ...
19
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9answers
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Learning Model Theory

What books/notes should one read to learn model theory? As I do not have much background in logic it would be ideal if such a reference does not assume much background in logic. Also, as I am ...
19
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5answers
7k views

In plain language, what's the difference between two things that are 'equivalent', 'equal', and 'identical'?

In plain language, what's the difference between two things that are 'equivalent', 'equal', 'identical', and isomorphic? If the answer depends on the area of mathematics, then please take the ...
18
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0answers
444 views

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 ...
17
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3answers
577 views

Applications of model theory to analysis

Some of the more organic theories considered in model theory (other than set theory, which, from what I've seen, seems to be quite distinct from "mainstream" model theory) are those which arise from ...
16
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1answer
691 views

Is there a natural model of Peano Arithmetic where Goodstein's theorem fails?

Goodstein's Theorem is the statement that every Goodstein sequence eventually hits 0. It is known to be independent of Peano Arithemtic (PA), and in fact, was the first such purely number theoretic ...
16
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1answer
730 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 ...
15
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2answers
373 views

A sentence false in a field of characteristic $0$ but true in all fields of positive characteristic?

Consider the language $L=\{+,\cdot, 0, 1\}$ of rings. It is easy to show using compactness that if $\sigma$ is a sentence that holds in all fields of characteristic $0$, there is some $N\in \mathbb N$ ...
15
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3answers
692 views

Comparing Category Theory and Model Theory (with examples from Group Theory).

The following question eats my brain: The standard definition of a "category" and a list of examples following this definition confuses me. My question is simple: (Q1)If someone write "the category ...
15
votes
2answers
632 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 ...
15
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1answer
988 views

(Why) is topology nonfirstorderizable?

Is it the right point of view to say, that topology is nonfirstorderizable (only) because the union of arbitrarily many open sets has to be open? And if "arbitrarily many" was relaxed to "finitely ...
15
votes
2answers
842 views

Intersection of Algebraic Topology/Geometry and Model Theory/Set Theory

Is there any intersection between the ideas of Algebraic Topology/Geometry (I know that there is most certainly a non-trivial intersection between Algebraic Geometry, Algebraic Topology, Arithmetic ...
15
votes
1answer
366 views

Is $\Bbb R$ definable in $(\Bbb C,0,1,+,*,\exp)$?

Is there a first-order formula ϕ(x) with exactly one free variable $x$ in the language of fields together with the unary function symbol $\exp$ such that in the standard interpretation of this ...
15
votes
1answer
131 views

Is it true that $\mathbb{C}(x) \equiv \mathbb{C}(x, y)$?

It is easily seen that any two consecutive entries in the tower of fields given below are not elementarily equivalent in the language of rings: $$\mathbb{Q} \subseteq \mathbb{Q}(\sqrt{2}) \subseteq ...
15
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1answer
402 views

A *finite* first order theory whose finite models are exactly the $\Bbb F_p$?

Since this question turned out to be trivial, I'm now asking this strengthened version: Is there a finitely axiomatized first order theory $T$ in the language of rings such that its finite models ...
15
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1answer
211 views

Tennenbaum's theorem without overspill

While trying to clean up Wikipedia's proof sketch for Tennenbaum's theorem (there is no computable non-standard model of Peano Arithmetic), the following strategy occurred to me. Since it seems to be ...
15
votes
1answer
364 views

What lessons have mathematicians drawn from the existence of non-standard models?

So, as someone whose knowledge of mathematics has always come from studying it with an eye towards philosophical/foundational issues and studying it with other philosophers (who are not primarily ...
13
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5answers
3k views

Example of non-isomorphic structures which are elementarily equivalent

I just started learning model theory on my own, and I was wondering if there are any interesting examples of two structures of a language L which are not isomorphic, but are elementarily equivalent ...
13
votes
3answers
2k views

Showing any countable, dense, linear ordering is isomorphic to a subset of $\mathbb{Q}$

I'm trying to knock out a few of the later exercises from Enderton's Elements of Set Theory. This problem is #17, found on page 227. A partial ordering $R$ is said to be dense iff whenever $xRz$, ...
13
votes
3answers
336 views

Axiom of Choice and Determinacy

In my set theory course we have been talking about the axiom of determinacy. One of the first things we showed was that $AD$ and $AC$ are incompatible. We later showed that $ZF+AD$ implies the ...
13
votes
4answers
2k 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 ...
13
votes
5answers
234 views

Does “=” have to be interpreted as equality?

To put it briefly: In model theory, we are allowed to interpret any relation symbol in any way we like. So why do people seem to require that "$=$" is interpreted as the actual equality? Let me ...
13
votes
3answers
470 views

On Pudlak's “Life in an Inconsistent World”

In his Logical Foundation of Mathematics and Computational Complexity (2013), Pavel Pudlak invites the readers to ponder about fictitious people whose natural numbers are nonstandard. His exposition ...
13
votes
0answers
468 views

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 ...
12
votes
3answers
700 views

Does every complete theory admit quantifier elimination?

Does every complete theory admit quantifier elimination? I know that at least in some simple cases the reverse is true; such as some reducts of number theory.Thanks
11
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6answers
739 views

Does a finite first-order theory which has a model always have a finite model?

I'm curious whether this is true or not: Let T be a finite, first-order theory. If T has a model, then T has a finite model. I would assume the answer is 'yes', but I wanted to make sure I ...
11
votes
4answers
1k views

How does Gödel Completeness fail in second-order logic?

So a while ago I saw a proof of the Completeness Theorem, and the hard part of it (all logically valid formulae have a proof) went thusly: Take a theory $K$ as your base theory. Suppose $\varphi$ ...
11
votes
2answers
498 views

Is algebra needed to really understand and/or enjoy model theory?

What are the desirable pre-requisites to be able to learn model theory well? In particular, it seems that connections to algebra are used heavily especially as examples. I would like to know if a ...
11
votes
2answers
679 views

Is Gödel's incompleteness theorem provable without any model-theoretic notion?

The entry on Gödel's incompletenss theorem in Wikipedia says: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, ...
11
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3answers
3k 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 ...
11
votes
1answer
446 views

First Order Logic: Prove that the infinitely many twin primes conjecture is equivalent to existence of infinite primes

I'm learning First Order Logic independently using a college textbook. I've been doing some self exercise question in it and came across this one, which I can't seem to figure out how to do: Let ...
11
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1answer
119 views

Surjective exponentials for algebraically closed fields

The existence of the exponential on $\mathbb{C}$ has a very basic, yet very strong consequence : $(\mathbb{C}^*,\cdot)$ is a quotient of $(\mathbb{C},+)$. This question is concerned with fields $K$ ...
11
votes
2answers
243 views

Putting down axioms for some symbols. Playing with their consequences qualitatively and symbolically. Building theories. The book?

I am interested in the design and building of theories. By building theories, I mean putting down axioms of various kinds, over various fields, exploring their perhaps interesting, or probably boring, ...
11
votes
2answers
340 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. ...
11
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3answers
382 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?
11
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1answer
296 views

Number of automorphisms of saturated models

I have the following assignment question: Let $M$ be an $L$-model of cardinality $\kappa$. Assume $M$ is saturated. How can you show that $|\text{Aut}(M)|=2^{|M|}$? I see two possible ...
10
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2answers
755 views

Is the compactness theorem (from mathematical logic) equivalent to the Axiom of Choice?

Or more importantly, is it independent of the axiom of choice. The compactness theorem states the given a set of sentences $T$ in a first order Language $L, T$ has a model iff every finite subset of ...
10
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5answers
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Most astonishing applications of compactness theorem outside logic

The compactness theorem has a lot of applications to logic and model theory. I'm looking for applications. I'm looking for theorems in other areas of mathematics which seem at first sight to have ...
10
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1answer
226 views

Do ordered fields and archimedian ordered fields have the same first-order theory?

Let us suppose that the first-order language of ordered fields has symbols for addition, subtraction, multiplication and order, and constant symbols for 0 and 1. An ordered field is said to be ...