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Question 1

1vote

1answer

111views

Complexity of $Th(\langle \mathbb{Z}; + , 1 \rangle)$ same as $Th(\langle \mathbb{Z}; + \rangle)$?

I want to know a lower bound for the complexity of the decision problem for $\langle \mathbb{Z}; + \rangle$. The below paper notes that Presburger arithmetic, originally $\langle \mathbb{N}; +\rangle$,...

Learner of math

287

asked2 days ago

Question 2

0votes

1answer

59views

Necessary and sufficient conditions for the theory of a quotient model to be equal to the theory of the main model

Given a language $\mathcal{L}$ and a model $\mathcal{M}$ in the language $\mathcal{L}$, what are the necessary and sufficient conditions that a congruence relation $\equiv$ must have such that $Th(\...

Diogo Santos

355

askedNov 25 at 16:25

Question 3

1vote

1answer

120views

Complete theory where we cannot omit all non-isolated types

The exercise is to give an example of a complete theory in a countable language such that no model of the theory omits all non-isolated types.I'm having trouble, especially since parameters are not ...

Anonymous Anonymous

99

askedNov 21 at 6:33

Question 4

5votes

2answers

159views

Construction of a dense linear order with $2^\kappa$ cuts

The question is the following:Given a cardinal $\kappa$, is there a dense linear order of size $\leq \kappa$ such that there are $2^\kappa$ cuts (a cut is a downward closed subset)?The question is a ...

Matteo Bisi

115

askedNov 20 at 11:20

Question 5

6votes

3answers

723views

Does truth not exist in ZFC, or is it merely not definable?

I am having a lot of trouble with the concept of Tarski's undefinability theorem as it relates to set theory.Tarski's undefinability theorem says that there is no formula $Tr$ on the natural numbers ...

William Oliver

1,262

askedNov 16 at 15:41

Question 6

2votes

0answers

81views

QE of $(\mathbb{N}, 0, 1, +, <, (\equiv_k)_{k \geq 2})$

I'm stuck on the proof for this. $(\equiv_k)$ is the binary relation for congruence modulo $k$.I began my proof by looking at a primitive formula $\eta(\bar{x}, y)$ and am attempting to find a finite ...

Anonymous Anonymous

99

askedNov 16 at 2:19

Question 7

0votes

1answer

148views

Relation between Tarski's conception of truth and (implicit) Axioms

I am trying to understand the original paper of Tarski Concept of Truth in the formalized languages, as printed in his collected works.I have read introductory texts from Shoenfield Mathematical ...

Alexander Wagner

49

askedNov 11 at 9:41

Question 8

2votes

1answer

116views

What real numbers correspond to second order logic definable Dekekind cuts?

This question spurred from a thought I had: does every (lower) Dedekind cut have a (finite) second order logic formula that defines it?Fix the usual setting: the domain is $\mathbb{Q}$ with the order ...

Markus Klyver

2,582

askedNov 7 at 21:51

Question 9

1vote

1answer

54views

two connected countable graphs elementarily equivalent but not isomorphic

I have been trying without avail. The theory of connected graphs is not definable of course.However, the theory of connected graphs with fixed diameter is definable, so I was thinking to somehow ...

Anonymous Anonymous

99

askedNov 3 at 1:22

Question 10

0votes

1answer

43views

elementarily equivalence implying isomorphism for countable

My professor proved Robinson's Joint Consistency Theorem and in one of the steps, took a theory in an expanded language with unary relations for structures and $W_n$ for $n \in \mathbb{N}$. He encoded ...

Anonymous Anonymous

99

askedNov 3 at 0:19

Question 11

10votes

0answers

362views

When is the iterated ultrapower innocuous?

Suppose $U$ is an ultrafilter over $I$. When does the isomorphism$$\prod_U \Big(\prod_U \mathfrak A_i\Big) \cong \prod_U \mathfrak A_i$$occur?I think I saw somewhere that if $U$ is a uniform $\...

edgar alonso

729

askedOct 30 at 23:41

Question 12

-2votes

1answer

98views

Get to know: The Tarski-Vaught Test [closed]

I want to understand the Tarski-Vaught Test and how to apply it.It is stated as follows.Let $\mathcal{M}$ ben an $\mathcal{L}$-structure and $A\subseteq M$. Then $A$ is the base set of an elementary ...

Window

49

askedOct 30 at 17:20

Question 13

1vote

2answers

74views

Finite Ehrenfeucht-Fraisse games with infinite constants

Suppose we have some theory $T$ on a language $L$ containing some (let's just say 1) relational symbol and a countably infinite number of constants. If we let $M,N$ be two models of $T$, define $G_n(M,...

Raymond Ying

147

askedOct 29 at 3:34

Question 14

4votes

1answer

135views

Why is the transfer principle embedding in nonstandard universes bounded?

I am trying to read Model Theory by Chang and Keisler. Given a set $A$, let $V(A)$ be$$\begin{align*}V_0 &= A \\V_1 &= V_0 \cup \mathcal{P}(V_0) \\\dots \\V_n &= V_{n-1} \cup \...

Link L

947

askedOct 25 at 12:03

Question 15

3votes

1answer

60views

Conditions for elementary submodels.

Here is a statement from my textbook: Let $L$ be a language and $M_1,M_2,M_3$ be $L$-structures. If $M_1 \preceq M_3, M_2 \preceq M_3$ and $M_1 \subseteq M_2$, then $M_1 \preceq M_2$, where $\preceq$ ...

ModelTheoryIsHard

551

askedOct 25 at 6:43

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Questions tagged [model-theory]

Complexity of $Th(\langle \mathbb{Z}; + , 1 \rangle)$ same as $Th(\langle \mathbb{Z}; + \rangle)$?

Necessary and sufficient conditions for the theory of a quotient model to be equal to the theory of the main model

Complete theory where we cannot omit all non-isolated types

Construction of a dense linear order with $2^\kappa$ cuts

Does truth not exist in ZFC, or is it merely not definable?

QE of $(\mathbb{N}, 0, 1, +, <, (\equiv_k)_{k \geq 2})$

Relation between Tarski's conception of truth and (implicit) Axioms

What real numbers correspond to second order logic definable Dekekind cuts?

two connected countable graphs elementarily equivalent but not isomorphic

elementarily equivalence implying isomorphism for countable

When is the iterated ultrapower innocuous?

Get to know: The Tarski-Vaught Test [closed]

Finite Ehrenfeucht-Fraisse games with infinite constants

Why is the transfer principle embedding in nonstandard universes bounded?

Conditions for elementary submodels.

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