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.
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Definability of the class of Hausdorff topological spaces using the closure operator in a first-order way.
Recall the Kuratowski closure axioms for general topological spaces:
$f(0)=0$, where $0$ is the empty set
$x \subseteq f(x)$
$f(x)=f(f(x))$
$f(x \cup y) = f(x) \cup f(y)$
The question I am ...
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what's means V is model of zfc?
I thought that set of formulas S has a model iff there is an interpretation satisfying that all the formula in S make true.
so I understand zfc's model is a kind of assignment that make all the zfc's ...
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Closure operators and uniqueness of the size of basis
Let $M$ be a set. A closure operator is a function $cl:\mathcal{P}(M)\to\mathcal{P}(M)$
such that
$\bullet$ $A\subseteq cl(A)$
$\bullet$ if $A\subseteq B$ then $cl(A)\subseteq cl(B)$
$\bullet$ $cl(cl(...
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Formula in upper Brackets Notation $\ulcorner \varphi \urcorner$: Meaning and Usage
I have some understanding problems with 2nd paragraph in following answer by spaceisdarkgreen:
[...] Note that from this perspective, when we write, e.g. $M\models \exists ! x\varphi(x)$ or $M\models ...
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Model unspecific Witness of an Existence Proposition
Let $T$ be some first order theory ( eg ZFC; feel free to assume this for the next), $\varphi(x)$ is a well- formed formula in $T$'s underlying language, and assume that the proposition $\exists^! x\, ...
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Definable types
Let $p(x)\in S({\mathcal U})$ be a global type. Let $\varphi(x,y)\in L$.
Assume there is a formula $\psi(y)$ such that $\psi(b)\Leftrightarrow \varphi(x,b)\in p$ for every $b$.
When is the case that $\...
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Existentially closed types [closed]
Is there an easy characterization of global types that are existentially closed?
A type $p(x)\in S({\mathcal U})$ is existentially closed if $\exists y\ \varphi(x,y)\in p$ implies that $\varphi(x,b)\...
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How to reduce to reduce to the case where the class $C$ is a set in this model theory exercise
I am working through the book "A course in model theory" by K. Tent and M. Ziegler. I am having issues with exercise 2.1 where we are given some class $C$ of $L$ - structures:
Let Th($C$) = ...
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Graduate Textbook Recommendations for Second-Order Logic [duplicate]
I'm currently reading Bruno Poizat's A Course on Model Theory and am interested in possibly extending model theory to second-order logic. I'm looking for recommendations on graduate-level textbooks ...
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Definable closure has the exchange property in an o-minimal structure
I'm having some trouble understanding the following proof from Macpherson's Notes on o-Minimality and Variations:
Theorem 2.2.2 [Pillay and Steinhorn 1986]. Let $\mathcal{M}$ be o-minimal, $A\...
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Can you give me an example of an implicit use of Godel's Completeness Theorem, say for example in group theory?
Wikipedia says that the more general form of Godel's completeness is "used implicitly, for example, when a sentence is shown to be provable from the axioms of group theory by considering an ...
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Is there a theorem that says we don't limit our notion of conservative extension by only focussing on set models?
From the current Wikipedia article on conservative extensions, it states:
an extension $T_2$ of a theory $T_1$ is model-theoretically conservative if $T_1\subseteq T_2$ and every model of $T_1$ can ...
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An Example of a Proof-theoretic Conservative Extension that's not a Model-theoretic One
Coming from here and the Wikipedia page on conservative extension, which defines a proof-theoretic conservative extension like this:
a theory $T_2$ is a (proof theoretic) conservative extension of a ...
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Understanding the witness property in the Henkin construction
I'm trying to get a really good intuition for why the proof of the compactness theorem via the Henkin construction in Marker's model theory uses the witness property, specifically, why use
For every ...
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If the Collatz conjecture is undecidable, then it is true
Suppose that the Collatz conjecture is undecidable in PA. Then, by Godel's completeness theorem, there are models where it is true, and models where it is false. Let M be a model where it is true. The ...
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Need help with a question about cardinality and perhaps model theory
Let $I$ be the unit interval. Call a collection $\Sigma$ of subsets of $I$ "valid" if it satisfies the following three conditions:
$I \in \Sigma$.
$\Sigma$ has the same cardinality as $I$.
...
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Can you define a topology so that all the parametrically definable subsets are open?
Let $\mathcal{L}$ be a first order language with only relation and constant symbols. Suppose $\mathfrak{A}=\langle A,\ldots\rangle$ is a structure for $\mathcal{L}$. Let $\mathscr{D}$ be the set of ...
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Large invariant types realized in $\cal U$.
Below $\cal U$ is a monster model; $\varphi(x;y)$ a given parameter-free formula; $a\in\cal U^y$.
When is the type $p(x)$ = { $\varphi(x;fa):f\in{\rm Aut}({\cal U})$ } realized in $\cal U$? (Assuming ...
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A first order theory of $\mathbb{R}$
The properties that $\mathbb{R}$ should satisfy can be summarized as "Dedekind complete ordered field".
The axioms of ordered fields are first-order sentences.
I came up with a first-order ...
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Difference between a Function and the name of a Function
I am currently studying the book by Loeb and Hurd on Nonstandard Analysis. In the chapter "simple languages for relational systems" they differ between an mathematical object like a function ...
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General theory of formal variable and algebra extension [closed]
Often in structure construction, a formal variable is introduced into a given structure (possibly with certain generating or constrained relation), resulting in a new structure. Are there any ...
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Is it possible to find a general algorithm for determining how many models for a given theory there are?
In this context there already exists a model and the theory is consistent. I'm leaning towards no, because of Turing's Halting Problem and the fact that you should be able to mimic a Turing machine ...
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Proof that any algebraically closed field is homogeneous.
Hi guys I'm currently try to solve one of the exercises in Marker's "Model Theory: An Introduction" and I'm kinda stuck. The exercise asks to prove that any algebraically closed field is ...
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Horn sentences and power structures
It is well-known that Horn sentences are "preserved" under products (see for instance Show that the direct product of structures satisfies a Horn sentence).
I was wondering what happens with ...
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The well-ordering number of second-order logic
The definitions and referred pages are from (Model-Theoretic Logics, Barwise 1985), primarily from chapter II. A free version is available here: Chapter II
Definition (pinning down ordinals): Let < ...
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"Pregeometry" on $\mathbb{F}_p$-Algebras
For a field $K$ of characteristic $p>0$, the operator $\text{cl}_K(A):=K^p(A)$ defines a pregeometry on $K$:
The conditions $A\subseteq\text{cl}_K(A)$, finiteness $\text{cl}_K(A)=\bigcup_{A_0\...
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Proof of a couple properties of maximally consistent sets of sentences
This is from p.10 of Chang and Keisler's Model Theory.
Lemma 1.2.10. Suppose $\Gamma$ is a maximally consistent set of sentences. Then: (i) for each sentence $\varphi$, exactly one of $\varphi$, $\...
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In what theory consistency proofs take place?
I am reading Kenneth Kunnen's Set Theory and had this question.
For instance, on Page 113 Lemma 2.3 states that: Assume $S,T$ are set of sentences in the language of set theory, and assume that for ...
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Axiomatization of hyperreal numbers
From wikipedia:
The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one ...
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For what theories are submodels closed for intersection?
The theories coming from a variety in the sense of universal algebra are a known example. Fields and ordered groups are important examples not covered by universal algebra. I also believe models of ...
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Exponential polynomials with infinite number of roots
An (ordered) exponential field is a structure $(F, +, \cdot, 0, 1, <, \exp^{(1)})$ such that $((F, +, \cdot, 0, 1, <)$ is an ordered field an $\exp$ is an isomorphism of ordered abelian groups $(...
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Formalizing a Theory in a Universe and Model vs Universe question
A basic question about a terminology on foundation of set theory: What does it mean to "formalize a formal theory (ie a collection of syntactically welldefined sentences with resp. to some ...
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Is ZFC's Wikipedia article accurate about the redundancy of the axiom schema of specification?
I have a question about the redundancy of the axiom schema of specification that mentioned in the Wikipedia article Zermelo–Fraenkel set theory. I am sure that in some formulations of ZFC, this axiom ...
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Metalanguage Integers
In context on set theory & model theory of set theory what does exactly mean "metalanguage integer(s)"? Recall a metalanguage ( where one reasons about object theory phrased in object ...
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How does the axiom of infinity allow us to conclude that $\{ 3,5,9 \}$ exists here? (Tao's Analysis I)
I am wondering why after defining the following axiom
Axiom 3.8 (Infinity). There exists a set $\mathbf{N}$, whose elements are called natural numbers, as well as an object 0 in $\mathbf{N}$, and an ...
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External Integers and First-Order Formulas in Set Theory
So far I know it is not possible to express the natural numbers $\Bbb N$ in terms of a first order sentence in language from ZFC. More precisely, one can in ZFC prove only it's existence as "...
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Is every Boolean algebra a subalgebra of a free one?
It is well-known - and indeed follows from usual algebraic facts - that every Boolean algebra is a quotient of a free one. Is every Boolean algebra moreover a subalgebra of a free one?
I believe this ...
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External vs Internal Properties for ZFC Models
I have a couple of questions about the notion of "external" vs "internal" in context of models of ZFC.
Following "generic" statement which says when one says Given a ...
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Internally Finite but Externally Infinite Set
I would like to understand the mechanism of following statement from this answer by Noah Schweber:
We could work in a non-$\omega$ model of ZFC. In such a model, there are sets the model thinks are ...
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Invariance under $\operatorname{Aut}(N / \{M\})$
Let $M\preceq N$ where $N$ is $|M|^+$ saturated.
Let $p(x)$ be a partial type over $\le|M|$ parameters.
What do we know (if anything at all) about when $p(N)$ is invariant under the action of $\...
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(When) are recursive "definitions" definitions?
This is a "soft" question, but I'm greatly interested in canvassing opinions on it. I don't know whether there is anything like a consensus on the answer. Under what conditions (if any) are ...
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Is the class of linearly-orderable rings first order axiomatizable?
A linearly ordered ring is a commutative ring $R$ with unity equipped with a linear order $\leq$ that is compatible with addition, and such that the set of nonnegative elements are closed under ...
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Does Löwenheim-Skolem require Foundation in any way?
As title states, I'm curious whether Löwenheim-Skolem (in either of its upward or downward versions) necessitates some implicit use of Foundation. The usual presentation makes quite clear the reliance ...
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Proving a simple consequence of the Compactness Theorem
I am self-learning logic, and trying to prove the following exercise using the Compactness Theorem:
Suppose $T$ is a theory for language $L$, and $\sigma$ is a sentence of $L$ such that $T \models \...
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Is quantifying over natural numbers non first order?
I was reading here that
Note that ‘x is an infinitesimal’ is not first order, because it requires you to quantify over the naturals.
Whats's non first order about quantifying over natural numbers?
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Is there a theory in which all types can be omitted?
Is there a natural example of a first order complete (consistent) theory $T$ in which every 1-type can be omitted? or is there always some isolated type? In that case, why?
Of course there are plenty ...
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Proving that the set of sentences that are true using the symbols $+,<,=$ is the same over all ordered fields
I am interested in whether the set of formulas that one can prove true for a concrete ordered field using the symbols $+,<$ and $=$, depends on the field. In particular, I am interested in the set ...
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in definition of assigment, what's means 'except possibly a'?
in frist-order logic,
part of assignments practice represent like this
"if 𝜙is ∀𝛼𝜓, where 𝛼 is a variable, then ⊨vℳ 𝜙 iff for every assignment 𝑣' that agrees with 𝑣 on the values of every ...
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Two families of isomorphic structures have isomorphic ultraproduct.
I am trying to prove the following result:
Let $(\underline{M}_i)_{i\in I}$, $(\underline{N}_i)_{i\in I}$ be two families of structures such that, for all $i\in I$, $\underline{M}_i \cong \underline{...
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Condition for dense isolated types [closed]
I need some help with proving the following:
Theorem. Let $T$ be a complete theory in a countable language and let $M \models T$.
If $|\mathcal{S}_n^\mathcal{M}| < 2^{\aleph_0}$ then the isolated ...
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