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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478 views

There is no smallest infinitely large prime

I'm reading Kunen's Foundations of Mathematics and trying to solve Exercise II.16.19, which constructs an elementary extension of the reals as an ordered field, and asks the reader to prove various ...
3
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2answers
100 views

Universe cardinals and models for ZFC

I'm reading through Joel David Hamkins' set theory lecture notes. On page 14, on the subject of inaccessible cardinals and submodels of ZFC in $V$, he defines a universe cardinal to be a cardinal ...
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1answer
47 views

How can we get rid of function symbols in model theory?

I understand the presentation of a language in logic as having relations with arities, functions with arities and constants. I understand that a constant can be thought as a function with arity $0$. ...
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1answer
70 views

What is the point of (Compactness Theorem in the) Overspill Principle?

I am trying to understand the basics of computation theory. The Overspill Principle (also at google) basically says if you are cool you can do everything Let Г be a sentence of predicate logic ...
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1answer
83 views

Existence of an automorphism of $\Bbb C$ that fixes a finite set pointwise but does not fix $\Bbb R$ setwise

I would like to construct a (ring-theoretic) automorphism of $\Bbb C$ that fixes a finite set $A$ pointwise but does not fix $\Bbb R$ setwise. Marker's Model Theory, Corollary 1.3.6 does that in this ...
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1answer
25 views

Partial order version of elementary equivalence

Elementary equivalence is an important concept in mathematical logic. Two models $\mathfrak{M}$ and $\mathfrak{N}$ of the same signature are elementarily equivalent, written $\mathfrak{M} \equiv ...
3
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2answers
54 views

Does having a filter which is not maximal implies the negation of Łoś theorem?

If we have a family $(\mathfrak{M}_i)_{i\in I}$ of $L$-structures, and a filter $\mathcal{F}$ over $I$, we can define the reduced product $\prod_{i\in I}\mathfrak{M }_i/\mathcal{F}$. If $\mathcal{F}$ ...
3
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3answers
89 views

Are surjective polynomial maps injective?

An injective polynomial map $p:\mathbb{C}^n\mapsto\mathbb{C}^n$ is surjective (Ax-Grothendieck theorem). What is known about the reverse implication (surjective implies injective)? Why does the ...
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1answer
74 views

Showing $2$ is not definable in $(\mathbb{Q},+)$.

As stated, I'm to show that $2$ is not definable in $(\mathbb{Q},+)$. I tried proving it by contradiction by showing that if $2$ were definable, then we could define $\mathbb{N}$ and multiplication ...
2
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2answers
83 views

Goedel's completeness theorem in and/or for intuitionistic first order logic

Warning: I am neither a logician nor a set theorist, just curious about foundations of classical and intuitionistic mathematics. Therefore it might well be that the things to come are plain wrong, and ...
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1answer
36 views

Prove that Th(N, +, .) has uncountably many 1-types over some finite set

not sure how to go about answering the above question. Thanks for your help! (alternative: Prove that Th(N, +, .) has uncountably many n-types over the empty set.)
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1answer
119 views

Is it possible to prove the axiom of infinity from the real number axioms?

What I mean by the title is whether, given that there is a class $\Bbb R$ and operations $+,\cdot,<$ that satisfy the ordered field axioms and the least upper bound axiom, you can prove the ...
3
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1answer
74 views

Showing there does not exist a formal proof of a formula $\phi$.

My problem:Suppose $R$ is a binary predicate and use the soundness theorem to show that there does not exist a formal proof of $$ \phi =\forall y\exists xR(x,y)\rightarrow \exists x\forall yR(x,y).$$ ...
5
votes
2answers
111 views

Finite Model Theory

It seems that finite model theory is regarded (in a sense) as a computer theoretic subject. Is this the case or are there questions of interest that are of interest to mathematical logicians or more ...
3
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1answer
85 views

“Syntactic models” and the proof of the Completeness Theorem

In Computational Complexity by Papadimitriou (page 107), he outlines the basic idea for a proof of the completeness theorem for first-order logic - namely, that given a consistent set $\Delta$ of ...
47
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8answers
4k 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 ...
2
votes
1answer
41 views

Non-isomorpic uncountable dense linear orderings (a textbook example)

We know that two countable dense open orderings are isomorphic, and we know that this is generally not true for uncountable structures. Now I quote from the textbook, that is "Model-Theoretic Logics": ...
3
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0answers
68 views

Finding a finite model

Hello I am having difficulty with this question, I am not even sure what strategy one would go about proving something like this: Suppose $L$ is a language which includes an infinite list ...
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2answers
69 views

Is there a name for models whose every element is named by (one or more) variable-free terms?

Let $T$ denote a first-order theory. Is there a name for those models $M$ of $T$ such that for all $x \in M$, there is a variable-free term in the language of $T$ whose interpretation under $M$ is ...
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3answers
138 views

ZFC Axioms to be extended?

Sorry if this is going to be a really loaded question. I was told several times that for virtually all theorems/corollaries/propositions of mathematics (except those cases not compatible with ZFC ...
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1answer
127 views

Definable subsets of the natural numbers using only the successor function

Consider the first-order language whose only nonlogical symbol is the unary function symbol $S$, and the structure $\mathfrak{N} = ( \mathbb{N} , S )$, where $S$ denotes the successor function. Why ...
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0answers
32 views

First-Order Logic: Non-Normal Model of Sentences True in all Normal Models?

Let $\mathcal{L}$ be a first-order language with quantifier $\forall$, connectives $\neg$ and $\rightarrow$, a two-place predicate $E$ and a one-place function symbol $f$. There are no other ...
3
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2answers
102 views

Model-theory and Proof-theory in Propositional Logic

I'm trying to link results of model theory and proof-theory in propositional language. Here i will use $\models$ to denote logical consequence, in the model-theory sense. Being $x,y$ two formulas of ...
2
votes
2answers
79 views

Which constructions on a category are still interesting for a groupoid?

By a groupoid, I mean a (small) category in which every morphism is an isomorphism. It looks to me that constructions on a category like the opposite ("dual") category or products and (co-)limits ...
3
votes
1answer
108 views

Is a homomorphism expected to be a (structure-preserving) map?

Is a homomorphism a special type of morphism, namely a structure-preserving map? For a morphism (of a category), it is clear that we can't always expect that a morphism is necessarily a ...
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2answers
230 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, ...
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1answer
80 views

Peano's theory of arithmetic and Gödel's 1st Incompleteness Theorem

Let $\mathcal{N}$ be Peano's 1st order theory of arithmetic and $\mathscr{A}$ it's standard model (which we assume exists). Infer from Gödel's 1st Incompleteness Theorem that there exists a closed ...
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1answer
33 views

Can I say something about the cadinality of this model?

Let $L$ be some first-order language. Suppose $A$ is existentially closed in $K$, a class of $L$-structures whose age is at most countable, and age($A$) is at most countable set . Can we say anything ...
2
votes
2answers
100 views

I would like some textbook recommendations for model theory

I am a 3rd year undergraduate math student and would like to study model theory. . I have some background with set theory, ordinals etc and also with mathematical logic. This is purely for self study ...
8
votes
1answer
125 views

Can all theorems of $\sf ZFC$ about the natural numbers be proven in $\sf ZF$?

I know a proof of Hindman's theorem that uses ultrafilters on the natural numbers, and ultimately, the axiom of choice. But the theorem itself is essentially a combinatorial property of the natural ...
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1answer
85 views

Example of two finitely isomorphic structures (or $\omega$-isomorphic) that are not partially isomorphic?

One can define several maps between given structures in order to portray similarities or differences. The concept of isomorphism displays a deep structural overlapping, while for instance an ...
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1answer
56 views

A Question About Tennebaum's Theorem?

Tennenbaum's theorem proves there are no countable recursive nonstandard models of Peano arithmetic. It is a proof by contradiction. If our countable, nonstandard model is recursive, then, given a ...
2
votes
1answer
50 views

Ax-Kochen theorem

I'm following the proof of Ax-Kochen theorem by this paper: http://arxiv.org/abs/1308.3897 I have two question in the proof of Ax-Kochen Principle: In the page 49, I didn't understend the ...
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vote
2answers
124 views

An exercise on (isolated) types

I am currently working through a model theory course and am doing some exercises from Marker's book. I am currently attempting exercise 4.5.2. Which states the following: Let $T$ be the theory ...
3
votes
1answer
114 views

Non forking extensions of types as extensions of filters

Given a set of parameters $A$ a type in $S_n(A)$ may be thought of as a maximal filter on the monster model which can be constructed from $A$-definable subsets. Given a type $q\in S_n(B)$ saying that ...
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vote
1answer
58 views

Show that every existential sentence is preserved upwards

A sentence is existential if it is of the form $\exists x_1 ... \exists x_nR$ and $R$ has no further quantifiers. A sentence is preserved upwards if and only if whenever it is true in an ...
4
votes
3answers
108 views

How are models constructed?

As I understand, a model $\langle S,\sim\rangle$ of $\mathsf{ZFC}$ is a set $S$ coupled with some binary operation $\sim$, in which the axioms of $\mathsf{ZFC}$ hold (please correct me if I am wrong). ...
0
votes
1answer
30 views

Prove $\{e\}$ is a model of group theory

In group theory one learns that there is exactly one trivial group of size 1, namely $\{e\}$. In addition to the axioms of the group theory, this group is uniquely determined by the axiom: DIM1 ...
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1answer
34 views

Definition of dicrete ordering

What means by "discrete ordering"?
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1answer
58 views

The definition of interpretation in a Kripke model collides with my intuition of what it should do

In Lindröm and Segerberg (2007) exposition of a Kripke model, with frame $F= \langle W,D,R,E,w_0\rangle$, they define an interpretation $I$ as a family of functions $I_w$, where $w$ ranges over ...
2
votes
1answer
41 views

Are all theorems of minimal arithmetic theorems of a given theory?

I am working on some metamathematics revision and the following question came up. Let the theory $R_0$ be axiomatized by the following axiom schemata which hold for all $n,m \in \mathbb{N}$: ...
0
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0answers
46 views

The unique model of cardinal $\kappa$ of a $\kappa$-categorical countable theory is saturated.

Let $T$ be a $\kappa$-categorical ($\kappa \geq \aleph_1$) first-order theory in a countable language $\mathcal L$. I try to prove that its unique (up to isomorphism) model of cardinal $\kappa$ is ...
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1answer
63 views

What is the definition of “Winning Strategy” in an Ehrenfeucht-Fraïssé game?

I've read many descriptions and applications of a Winning Strategy, as much as many for a Strategy tout court, but when a formal, algebraic definition is called upon, I've found close to no input. I ...
0
votes
1answer
60 views

What is the use of Tarski-Vaught test?

As title says, what is the use of Tarski-Vaught test? I do understand that it is necessary and sufficient criteria for $N$ to be an elementary substructure of $M$, but beside that I don't see how this ...
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2answers
34 views

What exactly is $L$-terms in model theory?

I got confused after seeing the inductive definition of $L$-terms in model theory. So I do get that there are variables and constants, and when function $f$ is applied to the term, the resulting thing ...
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1answer
41 views

Clarification regarding inner model, standard model, transitive model and Mostowski

After reading books before lectures, here's my thought regarding inner models and so on. Correct me if I am wrong. So there's universe $V$, which we assume to be the true universe. By Gödel's ...
5
votes
2answers
68 views

Removing sets from models of set theory

I have a naive and open-ended question: How can one remove a set from a model of set theory in such a way that the result is again a model of set theory? Directly related: what kinds of sets can ...
0
votes
1answer
61 views

A finite set of wffs has an independent equivalent subset

This seems to be a common exercise among many books (cf. Enderton, p. 28, van Dalen, p. 45, Hinman, p. 51, Chang and Keisler, p. 18), with some minor variations among them. The idea is simple. Say ...
2
votes
1answer
43 views

If $\phi$ is satisfiable and $\mathscr{S}$ is countable, then the set of all models of $\phi$ has the cardinality of the continuum

I have just started reading Chang and Keisler and I'm already stuck in an exercise. Let $\mathscr{S}$ be a countable set of sentence letters (i.e. $\mathscr{S} = \{S_0, S_1, S_2, \dots\}$ or some ...
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1answer
89 views

Is there a first order theory for equivalences classes?

Question will be a bit naive, so please, be kind. Consider a first order theory, $\Gamma$ . Let $\mathcal{M}$ be the category of models for $\Gamma$. Consider $\sim$ an equivalence relation on ...