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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7
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6answers
565 views

In what sense of “structure” do group homomorphisms “preserve structure”?

It is commonly said that group homomorphisms "preserve the structure of the group", e.g., from Wikipedia: The purpose of defining a group homomorphism as it is, is to create functions that ...
31
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 ...
13
votes
1answer
822 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 ...
5
votes
2answers
271 views

What axioms need to be added to second-order ZFC before it has a unique model (up to isomorphism)?

What axioms need to be added to ZFC2 (second-order ZFC) before the theory has a unique model (up to isomorphism)? I was thinking: adjoin the generalized continuum hypothesis (GCH) and a statement ...
9
votes
2answers
581 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 ...
11
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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 $xRz$, ...
7
votes
2answers
353 views

Number of Non-isomorphic models of Set Theory

Assume that the meta theory allows for model theoretic techniques and handling infinite sets etc (The meta theory itself is, informally, "strong as ZFC"). Also assume that I'm studying ZFC inside this ...
3
votes
1answer
629 views

$2\mathbb Z$ is not a definable set in the structure $(\mathbb Z, 0, S,<)$

This is (a translation of) an excerpt from a model theory textbook that shows that $2\mathbb Z$ is not a definable set in the structure $(\mathbb Z, 0, S, <)$, where $S$ is the successor function. ...
16
votes
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 ...
14
votes
2answers
559 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 ...
8
votes
0answers
270 views

Gödel's Completeness Theorem and logical consequence

At the end of a long process of "rumination" on "old" math log textbooks, I've found the "missing link" - from my personal point of view - between some issues I've raised in the previous months : (i) ...
9
votes
1answer
973 views

Show that binary < “less than” relation is not definable on set of Natural Numbers with successor function

After reading about the question, I've come to believe that it would suffice to exhibit and automorphism of that is not order preserving. However, I'm unsure of how to construct such an automorphism, ...
9
votes
2answers
326 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 ...
3
votes
2answers
215 views

dense linear orders DLO

I am asked to prove that if I have two models of dense linear orders DLOs WITH the minimum and maximum. must be izomorpic to each other by fining direct izomorphy. I seem to always get stuck ...
5
votes
2answers
919 views

A first order sentence such that the finite Spectrum of that sentence is the prime numbers

The finite spectrum of a theory $T$ is the set of natural numbers such that there exists a model of that size. That is $Fs(T):= \{n \in \mathbb{N} | \exists \mathcal{M}\models T : |\mathcal{M}| =n\}$ ...
2
votes
1answer
130 views

Show that if U is a principal ultrafilter then the canonical inmersion is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$

Show that: If $U$ is a principal ultrafilter, then the canonical inmersion $j$ is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$
1
vote
1answer
119 views

Why is class of one-dimensional vector spaces not axiomatizable?

Prove, class of all one-dimensional vector spaces over R isn't axiomatizable in the signature σ=< +, α, 0>, α* is single function vector multiplication by a scalar $\alpha$ of R. There are ...
0
votes
2answers
134 views

Union of definable sets

I tried to prove this question but without a success: Let $K_1$ and $K_2$ be definable sets, prove that $K_1\cup K_2$ is definable. What I tried to do is to assume: $K_1=Ass(X)=\{ v|v \vDash X \}$ ...
15
votes
9answers
1k views

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 ...
14
votes
3answers
659 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 ...
8
votes
5answers
840 views

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 ...
22
votes
1answer
409 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 ...
16
votes
3answers
442 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 ...
7
votes
1answer
586 views

What's the best way to teach oneself both Category Theory & Model Theory?

I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So ...
6
votes
1answer
360 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 ...
4
votes
2answers
519 views

Model existence theorem in set theory

From the FOM newsgroup I learned: It's a theorem of (first-order) set theory that every consistent first-order theory has a model. What's the exact formulation of this theorem in purely ...
11
votes
4answers
1k 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 ...
9
votes
2answers
356 views

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 ...
6
votes
3answers
539 views

Complete first order theory with finite model is categorical

I am trying to prove that if $T$ is a complete first order theory that has a finite model then it has exactly one model up to isomorphism. To this end, I assumed that $T$ is complete with a finite ...
5
votes
1answer
418 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 ...
3
votes
1answer
307 views

Quantifier Elimination

I want to prove the following: The structure $(\mathbb{Z},\equiv,0)$ has QE (with $\equiv$ a relation such that for all $m,n\in\mathbb{Z}$: $m\equiv n$ iff $m-n$ is even). I thought hereover in the ...
1
vote
3answers
208 views

A quick question about categoricity in model theory

I just want to see if I am using the term "categoricity" correctly in the following context: (1) I was thinking about why someone might reject a simple resolution to Skolem's Paradox. (2) The ...
6
votes
2answers
199 views

Model theory in group theory

I am interested in useful results for group theorists that can be shown using model theory. For example : Theorem: Let $\langle X \mid R \rangle$ be presentation of a group $G$ with $X$ finite and ...
6
votes
2answers
234 views

Boolean algebras without atoms

Why is the theory of Boolean algebras without atoms $\omega$-categoric?
5
votes
1answer
229 views

In ZF, does there exist an ordinal of provably uncountable cofinality?

Question is in the title. In ZFC, one can prove that $\aleph_{\alpha+1}$ is regular, so there is a large source of cardinals with uncountable cofinality, but in ZF, it is consistent that ${\rm ...
3
votes
2answers
226 views

how to create the set of hyperreal numbers using ultraproduct

As title says, can anyone explain how to create the set of hyperreal numbers using ultraproduct from the set of real umbers?
2
votes
3answers
270 views

Weak categoricity in first order logic

In a certain sense, only finite structures are definable up to isomorphism in first order logic. But if we rely on a metatheory containing a sufficient strong set theory (like required for second ...
5
votes
3answers
263 views

Confusion of the decidability of $(N,s)$

In some context the PA has only the successor operator $'s'$, but in logic we always refer the structure of PA is $(\mathbb{N},0,1,s,+,\times)$. I believed the theory of the two sturctures are ...
4
votes
1answer
93 views

Models of the empty theory

so throughout my reading of model theory the idea of the "empty" theory has been put down as trivial, however I am curious as to why. Let us look at the following. Suppose We have $L_=$, the language ...
4
votes
1answer
230 views

Types and Algebraicity

Let $\Phi(\bar x)$ be a type over a set $X$ with respect to a structure $A$. Show that if $\Phi$ is algebraic, then $\Phi$ contains a formula $\phi$ s.t. $A\models\exists\ _{<n}\bar x\phi(\bar x)$ ...
3
votes
1answer
144 views

Automorphism of an elementary extension of a structure that moves an undefinable element

I know that the easiest way to show a point is not definable is to find an automorphism of the structure that moves the given point. I've also seen many examples undefinable points that couldn't be ...
3
votes
1answer
130 views

Existence of elementary substructures of a uncountable structure over a countable language

Let $\kappa$ be an regular uncountable cardinal carrying a $\tau$-structure for some countable language $\tau$. What can be said regarding the existence of ordinals $\alpha <\kappa$ carrying ...
3
votes
1answer
153 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 = ...
2
votes
1answer
156 views

The connection between quantifier elimination, $\omega$-categorical and ultrahomogenous

A relational structure $A$ with an $\omega-$categorical theory $Th(A)$ is ultrahomogenous iff $Th(A)$ admits quantifier elimination. (*) I was wondering wether the structure $A$ has to be ...
2
votes
1answer
221 views

Does elementary embedding exist between two elementary equivalent structures?

By previous question, if there is a elementary embedding from $\mathfrak A$ into $\mathfrak B$, then $\mathfrak A \equiv \mathfrak B$. Now it is naturally to ask conversely, if $\mathfrak A \equiv ...
2
votes
2answers
75 views

Confused about models of ZFC and passage of book

I have a question about a passage of Just/Weese. In the following, let $\mathcal M = \langle M, E \rangle$ be a model of ZFC. Here is the first half of what I'm about to ask a question: So I asked ...
1
vote
1answer
340 views

Is one structure elementary equivalent to its elementary extension?

Let $\mathfrak A,\mathfrak A^*$ be $\mathcal L$-structures and $\mathfrak A \preceq \mathfrak A^*$. That implies forall n-ary formula $\varphi(\bar{v})$ in $\mathcal L$ and $\bar{a} \in \mathfrak A^n$ ...
27
votes
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, ...
22
votes
1answer
620 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 ?
19
votes
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 ...