Questions tagged [model-theory]
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 languages. Not to be confused with mathematical modeling.
4
votes
0answers
70 views
How is it possible to have a model of a set theory? [duplicate]
I am trying to understand the basics of model theory. Before getting too deeply into it, I would like to know how it is even possible to construct a model, i.e. a structure that satisfies axioms of a ...
2
votes
1answer
84 views
What is the purpose of Semantics/Model theory in Mathematical Foundations?
First off I know very little model theory so apologies if I say anything very dumb or offensive to logicians/model theorists. Second I should note that a lot of what I am saying here is motivated by ...
0
votes
0answers
37 views
Amalgamation base, extending Galois type
Here on the page 12 the Observation 1.11 5) says
If $M\leq_{\frak K} N$ are from ${\frak K}_{\lambda}$, $M$ is an amalgamation base
and $p\in \mathscr{S}(M) \;\underline{\text{then}}$ there is $q\in ...
1
vote
0answers
43 views
Why is a Skolem theory model complete?
Let $\Delta$ be a Skolem theory. Let $M$ be a model of $\Delta$ and $N$ a substructure of $M$. Then we need to proof that for every $L_N$-sentence $\phi$ we have the equivalence $N \models \phi \iff M ...
0
votes
0answers
34 views
Fixing and pointwise-fixing of a structure in a suqare
Suppose we have this square in the context of Abstract Elementary Class ${\frak K}=(K,\leq_\frak K)$:
$$
\require{AMScd}
\begin{CD}
N_1 @>f_1>> N_3 \\
@A{\preceq_\frak K}AA @AA{f_2}A\\
M_0 @&...
3
votes
1answer
67 views
Ultraproduct of simple structures
Definition.
1) A complete first-order $\mathcal{L}$-theory $T$ is said to be simple if each type does not fork over some subset $A$ of its domain where $|A|\leq |T|$.
An $\mathcal{L}$-structure $\...
0
votes
2answers
44 views
Question about $\Gamma_\infty^M$ deriving a very specific wff
In p.69 of Tourlakis's mathematical logic book.
He pulls a very specific theorem seemingly out of thin air.
$\Gamma_\infty^M \vdash \exists_x(f \bar n ... = x)$
I'm not sure how this is derived as ...
0
votes
1answer
42 views
Elementary equivalence of standard and non-standard model of arithmetic
There is the common construction of a non-standard model of arithmetic by adding a constant symbol c to the signature and adding $\{n<c:n\in \mathbb{N}\}$ to the theory PA. Now by adding all ...
-1
votes
0answers
32 views
Base enlargement/isomorphism of models over a model [closed]
How can I see that if models $M_0$ and $M_1$ are isomorphic over $N_0$ and $N_0\leq_{\frak K} N_1$ then they are isomorphic over $N_1$? It is almost obvious but not quite.Any simple argument will help....
1
vote
1answer
44 views
Why does the compactness theorem not apply to infinite languages?
I am trying to show that compactness doesn't apply to infinite languages like $L_{\omega_1,\omega}$ that allow infinite FOL sentences like $\forall x\, \bigvee_{n\in \omega} x \approx S^n(0)$, i.e. $\...
1
vote
1answer
62 views
Proving that $\langle\mathbb{R}-\{7\},<\rangle$ is an elementary substructure of $\langle\mathbb{R},<\rangle$
Prove that $\langle\mathbb{R}-\{7\},<\rangle$ is an elementary substructure of $\langle\mathbb{R},<\rangle$.
What I thought I should do is to use induction on statements to prove that. Any ...
1
vote
1answer
56 views
What does this set-theoretic notation mean in the proof of this theorem?
From A friendly introduction to mathematical logic,
In the first red box below, what does this notation mean?
And in the second red box, how exactly is this process repeated?
0
votes
1answer
53 views
Can we have models that do not satisfy unproved existential sentences in the theories they model?
For any first order theory $T$, let's say that $M$ is an ideal existential model of $T$ if and only if, for every formula $\varphi(x)$ in the language of $T$ in which only the symbol "$x$" appears ...
2
votes
2answers
61 views
The theory of dense linear orders without end-points is not $2^\omega$-categorical
It seems best to prove this by counter example. Both $\mathbb{R}$ and $I := \mathbb{R} \backslash \mathbb{Q}$ under the usual order $<$ are models of the theory of dense linear orders without end-...
4
votes
2answers
183 views
Prove that all algebraic numbers are included in any elementary substructure of $\mathbb R$
Let $A$ be an elementary substructure of $\mathbb R$ where $\mathbb R$ is $\langle \mathbb R,+,\cdot,0,1\rangle$ . Show that $A$ contains any algebraic number.
What I tried to do was use the fact ...
3
votes
1answer
36 views
Reduct of a first-order structure
I am trying to understand the notion of reduct in model theory. I found the following definition in Hodges's model theory book.
Definition. Let $\mathcal{L}$ be a first-order language and $\mathcal{L}...
6
votes
1answer
117 views
How to give formal proof in logic?
I am taking a course on mathematical logic and am struggling to give formal proofs of theorems or claims. Currently doing first order logic. Here, an example:
Claim: $\models (\exists x)\big(A(x)\...
2
votes
2answers
66 views
Why is it that the complete L-theories containing $\Sigma$ are exactly those of models of $\Sigma$?
Consider a set of L-sentences $\Sigma$ and a set of complete L-theories $T_i$ containing $\Sigma$ i.e. $\Sigma \subseteq T_i$.
Why is it that for each complete L-theory $T_i$ containing $\Sigma$ are ...
2
votes
1answer
55 views
Criterion for $\forall\exists$-axiomatizability.
I am stuck in following exercise:
Let $T$ be a theory with following property: for all models $M$, $N$, and $P$ of $T$ if $M \subset P$, $N \subset P$, and $M \cap N \neq \emptyset$, then $M \cap N ...
4
votes
2answers
86 views
how a theory can be categorical of a large cardinality?
ehrenfeucht mostowski models eliminate types,also we can find saturated models of arbitrary large cardinalities, so I got confused how a theory can be categorical of that large cardinality?
0
votes
0answers
56 views
Boolean algebras/Unknown notation
Does someone know what is meant (in the context of trees and Boolean algebras by Shelah) here on the page 8 right above Remark 1.5:
$$\{\langle\rangle\}\cup\{\langle\xi\rangle\otimes_{\zeta(*)}d\eta:\...
3
votes
0answers
54 views
Chang & Keisler Exercise 4.3.9
I'm having a lot of trouble sorting out an exercise from Chang & Keisler's Model Theory:
4.3.9: Let $I$ be an infinite set of power $\alpha$. If $E \subset P(I)$, $|E| \leq \alpha$, and the ...
0
votes
1answer
18 views
Conjunction closure of positive primitive formulas
Let $\mathfrak M = (M, 0, +, - , r)_{r\in R}$ be a structure which realizes a module, i.e., $(M, 0, +, -)$ is an abelian group together with operations $r : M \to M$ with $R$ a ring with $1$. Let $L_{...
0
votes
1answer
39 views
Intuitive understanding of definitional expansion in model theory
I was trying to understand what definitional expansion mean. Consider having two L-structures $\mathcal A$ and $\mathcal A'$ where $A'$ is an L'-expansion of $\mathcal A$. We way that $\mathcal A'$ is ...
0
votes
0answers
28 views
Why is it if every model of $\Sigma$ has an L'-expansion then $\Sigma'$ is conservative over $\Sigma$?
Consider $\Sigma \subseteq \Sigma'$ and $L \subseteq L'$. Then the theorem is:
If every model of $\Sigma$ has an L'-expansion, then $\Sigma'$ is conservative over $\Sigma$.
Recall conservative ...
2
votes
0answers
42 views
Natural models of Ackermann Set Theory
Consider Ackermann's Set Theory (as described here, §6, including the axiom of regularity for sets), henceforth denoted AST, as a theory in the language $\left\langle \in,\mathbf{V} \right\rangle$ ...
1
vote
1answer
77 views
The cornerstone definition in Abstract Elementary Classes
In the first paragraph of p.41 of Introduction to: classification theory for abstract elementary class, Shelah gives the following definition of (Galois) type.
For $M\preceq N_\ell$ and $a_\ell\in ...
0
votes
1answer
66 views
Q: On Soundness proof in Tourlakis 2003
In p.59 of Lectures in Logic and Set Theory, Tourlakis proves the inductive step for the E-introduction rule of Soundness.
He uses a proof by contradiction stating that $B'[x \leftarrow \bar i] \...
1
vote
1answer
24 views
Sentence over the empty language saying a model is infinite
So the question is whether there is a sentence $\phi$ over the empty language such that for any structure $M$, $M\models\phi$ iff $M$ is infinite.
I’m pretty sure the answer is no, but I don’t see ...
0
votes
2answers
56 views
A theory $T$ is model-complete if the union of $T$ with an atomic diagram is complete
Let $T$ be a theory in first order logic over some language $L$. Let $\mathfrak A$ be some structure over $L$ with $\mathfrak A \models T$ and with $A$ be its universe. Then consider every $a \in A$ ...
0
votes
0answers
39 views
What is the relationship between the common concept of “model” and “model” as used in Model Theory? [duplicate]
To my understanding, a model in Model Theory is an interpretation (in a form of a set or other algebraic structures) for a certain sentence S which makes S true.
In everyday language, and also in ...
1
vote
1answer
30 views
Notion of submodel relation
There is no definition of the essential notion of substructure (=submodel) in Shelah's introduction E56 to AEC, 1st Volume.
Could someone please define this for me?
I think that
$$M \subseteq N$$ ($M$ ...
2
votes
1answer
57 views
The structure $(\mathbb N, <, f)_{f\in \mathcal F}$ seems to have no elementary extension
This is exercise 2.3.1 (2) from Tent/Ziegler, A Course in Model Theory.
Let $\mathcal F$ be the set of all functions $\mathbb N \to \mathbb N$. Show that $(\mathbb N, <, f)_{f\in \mathcal F}$ ...
1
vote
1answer
54 views
Find non-isomorphic models of $(Q,<,c_{n \in N})$
This is a problem in Basic Model Theory by Kees Doets:
Let
$X=(Q,<,n)_{n \in N}$
$Y=(Q,<,\frac{-1}{n+1})_{n \in N}$
$Z=(Q,<,q_n)_{n \in N}$ where $\{q_n\}_{n \in N}$ is an ascending ...
3
votes
1answer
68 views
Existence of saturated models of a theory.
In Tent, Ziegler: A Course in Model Theory it is stated on page 89, that
Saturated structures need not exist (think about why not), but
by considering special models instead, we can preserve many ...
3
votes
1answer
32 views
Isomorphic subsets of countable total orders
Suppose $(\Omega,\leq)$ is a totally ordered set, with $\Omega$ infinite and countable. If $S$ is an infinite subset of $\Omega$, then $(S,\leq)$ denotes the induced totally ordered set.
Are there ...
7
votes
0answers
180 views
Does naturality imply independence from choice?
My question concerns the notion of naturality defined in Section 12.3 of Hodge's (longer) Model Theory. See the addenda for the definition.
Hodges proves a result that if an algebraic construction ...
3
votes
1answer
62 views
Let $L = \{<\}$ prove there is no $L$-theory that has precisely the well ordered sets as models
This is the literal question: let $L$ be the language that only contains the two place relation symbol $<$. Prove there is no $L$-theory that has precisely the well ordered sets as models.
I ...
0
votes
0answers
39 views
What do sentences in the theory of the structure $A=(Q,<,n)_{n \in N}$ look like?
I'm working on a problem from Kees Doets, and he mentions the following structures:
$X=(Q,<,n)_{n \in N}$
$Y=(Q,<,\frac{-1}{n+1})_{n \in N}$
$Z=(Q,<,q_n)_{n \in N}$ where $\{q_n\}_{n \in ...
2
votes
1answer
38 views
model theory - existence of a model
Let $\Gamma$ be a consistent set of $L$-sentences with infinite cardinality. If it has an infinite model, then there exists a model for
$$
\Gamma' =\Gamma \cup \{\lnot c_a = c_b : a \neq b \},
$$
...
1
vote
3answers
57 views
If $\Sigma \models \phi$, then for some finite $\Delta \subset\Sigma$, $\Delta \models \phi$.
This is an easy consequence (Doets calls it Compactness Theorem (version 2)) in Kees Doets' Basic Model theory:
Let $\Sigma$ a set of sentences and $\phi$ a sentence.
If $\Sigma \models \phi$, then ...
0
votes
0answers
52 views
Boolean algebras/ Shelah/ Unclear step in the proof
Here on the page $10$, what does the displayed formula in the $6$th line
$$\text {lg}(\eta_\ell)<\omega\Rightarrow \bigcup\{\text{Rang}(\nu_\ell(k):k<\text{lg}(\nu_\ell))\}\cap\bigcup_{k<\...
0
votes
1answer
35 views
model theory - completeness of a theory
For an $L$-structure $\mathcal{A}$, the $L$-theory of $\mathcal{A}$ is the set of $L$-sentences:
$$
\mathrm{Th}_L(\mathcal A) = \{\sigma : \mathcal A \models \sigma\}
$$
Prove that $\mathrm{Th}...
1
vote
2answers
40 views
Craig interpolants for inifinite set of implications
Suppose we have an infinite set of first-order sentences $\{\alpha_i\}_{i=1}^{\omega}$ and first-order sentence $\beta$ such that for all $i$,
\begin{align*}
\alpha_i \vDash \beta
\end{align*}
I ...
1
vote
1answer
64 views
What is the difference between the logical formula $\alpha$ and expression “$\alpha$ is true”?
What is the difference between the logical formula $\alpha$ and expression "$\alpha$ is true"? I feel that there is a difference between those expressions. I think this difference is the difference ...
2
votes
1answer
25 views
Ultraproducts with a non-principal ultrafilter is aleph-one compact
I came across the following statement:
Given an $\omega$-sequence of $\mathcal{L}$-structures, $(\mathcal{M}_i : i < \omega)$, and a non-principal ultrafilter $\mathcal{U}$ on $\omega$, the ...
1
vote
1answer
36 views
Prove that the topological closure of a set is definable if the set is definable
Let $L$ be the language $L=\{<,=,+,-,\cdot, 0,1\}$, with standard interpretations, and let $\mathcal{A}=\langle\mathbb{R}, <,=,+,-,\cdot,0,1\rangle$. Let $S\subseteq\mathbb{R}^n$. Show that if $...
1
vote
0answers
90 views
A model of $PA^\omega$ in ZFC [closed]
It is known that the existence of a model (i.e. consistency) of $PA$ is provable in ZFC, the set $\omega$ equipped with usual operation on natural numbers is indeed a model of $PA$, and we can think ...
0
votes
2answers
44 views
Logic problem about set of disjunction forms(Fixed)
I will call the disjunction of literals a disjunction form. Let $\Sigma$ be a set of disjunction forms, and $\alpha$ is a well-formed formula satisfying $\Sigma \vDash \alpha$. For all propositional ...
2
votes
1answer
70 views
Dimension of vector spaces under surjection
Definition. Let $V$ be a vector space and $X\subseteq V$ be a definable subset in the language of vector spaces. $X$ is independent if for all $a\in X$, $a\notin span(A\setminus\{a\})$.
Definition. ...
