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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Intuition behind downwards Lowenheim-Skolem Theorem
I understand the formal proof of the Downwards Lowenheim-Skolem Theorem, and could probably reproduce it if I were asked to, but I'm not so certain about the intuition behind the proof and what we're &...
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Meta-logical terminology
What do we call a theory $\Gamma$ when...
For every sentence $\varphi$, either $\Gamma\vdash\varphi$, or $\Gamma\vdash\neg\varphi$?
There is no sentence $\varphi$ such that both $\Gamma\vdash\varphi$ ...
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Model theory of intuistionistic logic
The Tarskian interpretation of first order logic, $I=I_{S,\alpha}$, where $S$ is a first order structure w.r.t. the signature $\Sigma$, and $\alpha$ is a variable assignment, assign either $T$ or $F$ ...
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How do you relativize a replacement axiom in ZFC?
I'm reading the section 'Natural Models' in Enderton's 'Elements of Set Theory' p 249. Theorem 9L states that if $\kappa$ is an inaccessible cardinal then all the axioms of ZFC are true in $V_\kappa$. ...
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Understanding an example of forcing
In Chapter 14 of Jech's Set Theory, example 14.2 is as follows:
I understand each individual step, but I don't really get what this is actually doing. It says something about "adjoining a new ...
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Why does forcing not obey some simple rules of Propositional Calculus?
In Cohen "Set Theory and the Continuum Hypothesis" Cohen states on page 118:
"Also forcing does not obey some simple rules of the propositional calculus. Thus p may force $\neg \neg A$ ...
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What algebraic structure can you give sets equipped with an infinite sequence of Boolean lattices?
Is there a name for a set equipped with an infinite sequence of Boolean lattices?
What kinds of structure can you give to this class?
For example, is there a natural notion of a product?
Let's talk ...
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Question on the proof of omitting types theorem
I am reading the proof of omitting types theorem (page 125 - 127) in Marker's Model Theory: An Introduction (the same proof is in https://proofwiki.org/wiki/Omitting_Types_Theorem) and having the ...
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Do we have a translation from intuitionistic logic to classical logic which can translate all formulas with their precise meaning?(not only theorems)
I want to know if there is a translation from intuitionistic propositional logic formulas to classical propositional logic formulas satisfying the properties I'm looking for.
Actually first part of my ...
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Are the field reducts of real closed fields finitely axiomatizable?
This is a follow-up to my previous question on real closed fields, here: Are the field reducts of real closed fields first-order axiomatizable?. In that question, I asked whether the $\{+,-,*,0,1\}$ ...
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Questions about proof of consistency of GCH
In Jech's Set Theory, Theorem 13.20 states the following:
If $V=L$ then $2^{\aleph_\alpha}=\aleph_{\alpha+1}$ for every $\alpha$.
In the proof, he writes the following:
Let $X\subseteq \omega_\...
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"Equivalence" between different structures over expansions of reals as an ordered field
In the beginning of the article "Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function", by AJ Wilkie, ...
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Can a model of $\sf ZFC$ be a subset of $V_\omega$?
Can a model of $\sf ZFC$ be a subset of $V_\omega$? If so, in which theory this is provable?
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The equality-free first-order theory of antisymmetric relations and of coreflexive relations
This is yet another of my questions regarding first-order logic without equality. It is actually two questions. First, some definitions: An antisymmetric relation on a set $S$ is a binary relation
$R$ ...
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Trakhtenbrot and monotonicity
This is an exercise I saw in the lecture notes about Finite Model Theory. https://courses.cs.washington.edu/courses/cse599c/18sp/hw/hw3-whitespace.pdf
Let $\varphi(x)$ be formula with a free variable ...
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An infinitely axiomatizable class of structures whose reduct is finitely axiomatizable, and vice versa
Let $L$ and $L'$ be first-order languages such that $L' \subseteq L$. Does there exist a class $K$ of $L$-structures which is axiomatizable but not finitely axiomatizable, such that the $L'$ reduct of ...
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Creating an ultrapower of V by an ultrafilter over V
I've been reading about ultrapowers, and one of the most interesting things about them is when you build them using various large cardinals. For instance, taking the ultrapower of $V$ by a measurable ...
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Can a transitive model of $\sf ZC$ satisfy false statements about modeling within it?
This comes in connection with this posting.
My question here is what happens if we only make the sole change of having the mother model $M$ satisfying $\sf ZC$ instead of $\sf ZFC$.
To clarify, form ...
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How to express a function $f:A\rightarrow B$ as a first order structure?
Suppose $A$ and $B$ are non-empty sets and $f:A\rightarrow B$ is an onto function
What is the standard way to express this as a first order structure?
For example, one may consider a language $L$ ...
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Is there an accessible $\omega$-model of this theory with $\sf ZFC$ inside $\sf ZC$?
I've once suggested the following theory as a theory that meets Muller's criteria: page 14 for a founding theory of Mathematics. I'll re-exposit it here:
Define an upward membership chain as a ...
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Can we have this sequence of transitive models of ZFC?
This comes in follow up of this posting.
Can we have a transitive model $M_\omega$
of $\sf ZFC$ in which there exists a sequence $(M_n)_{n \in \mathbb N}$ of transitive models of $\sf ZFC$ such that $...
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Is this infinite sequence of transitive models of ZFC equiconsistent with Con(ZFC)?
Is it provable in $\sf ZFC$ that the existence of a transitive model of $\sf ZFC$ implies the existence of a sequence $(M_n)_{n \in \mathbb N}$ of transitive models of $\sf ZFC$ such that $M_m \in M_n$...
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I need some help with the syntax of David Marker's introduction to Model Theory
I seem to have gotten the hang of most of his syntax in the early chapters, but I need help/translation with a few details
For
ii) If $s$ is the variable, $v_{ij}$, then $s^M(\overline{a}) = a_{ij}$
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What are supervaluations?
What are supervaluations?
I'm interested in how you would define these things formally for first-order free logic.
In particular, I'm interested in whether the collections of extensions are a set or a ...
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Are the field reducts of Euclidean ordered fields an axiomatizable class?
An Euclidean field is an ordered field $(F;+,-,*,0,1,\leq)$ such that every positive element has a square root. So, Euclidean fields are a first-order axiomatizable class. But, what about the field ...
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Witnessing expansion of a first-order language and the use of the Axiom of Choice to expand any structure in the language
Understanding the use of the Axiom of Choice in the constructions mentioned in this question prompted the following, which I will state as a Lemma:
Lemma: Let $\mathcal{M}$ be any $\mathcal{L}$-...
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Understanding the use of the Axiom of Choice in David Marker's proof of Compactness
I am trying to understand the use of the Axiom of Choice in the proof of Lemma 2.1.8 (page 37 below) in David Marker's Model Theory: An Introduction
Pages 35 and 36 can be found here. Lemma 2.1.8 is ...
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How does formalization work in mathematics?
I would be extremely grateful is someone could review/comment/complement my reasoning and understanding of formalization in mathematics.
Let $T$ be a mathematical theory, say real analysis. $T$ is ...
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Is union of family of inner models also an inner model?
For every ordinal $\alpha$ let $M_\alpha$ be an inner model of $\mathsf{ZF}$ so that for every set of ordinal $S$ there exists an ordinal $\beta$ such that $\bigcup_{\alpha \in S} M_\alpha \subset M_\...
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Cardinality of Dedekind Completion of Hyperreals
Let $^*\mathbb{R}$ denote the hyperreal field constructed as an ultra power of $\mathbb{R}$. At the expense of losing the field properties, we may take the Dedekind completion of $^*\mathbb{R}$ to get ...
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Does there exist a $\lambda$-good, $\omega$-incomplete filter on $\kappa$ such that $P(\kappa)/\lambda$ is $\lambda$-saturated?
Let $\kappa$ and $\lambda$ be cardinals (we can assume $\lambda < 2^\kappa$). Does there exist a $\lambda$-good, $\omega$-incomplete filter on $\kappa$ such that the quotient $P(\kappa)/\lambda$, ...
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How to construct ZFC from scratch?
I am interested in understanding the foundations of mathematics. This naturally leads to the study of set-theory and its different axiomatizations. My issue is with the different ways to construct ...
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Number of models of the naturals and reals with and without CH
The Wikipedia page on true arithmetic says that it has $2^\kappa$ models for each uncountable cardinal $\kappa$. This refers to the theory of all first-order statements of the naturals.
I'm curious ...
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Inequality of Morley Ranks
I'm reading theses notes on model theoretic stability and I got to an exercise on p. 43:
Show that if $X \subseteq \mathbb{M}_{x}, Y \subseteq \mathbb{M}_{y}$ and $f: X \rightarrow Y$ are definable, ...
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Soundness and Completeness for a single Model Only?
Question modified to hopefully answer the questions (I'm a physicist to all might not be mathematically watertight)
In Enderton "A Mathematical Introduction to Logic", logical Implication is ...
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Downsides of defining $\models$ to be false when there's a mismatch in signature between the structure and formula.
The way that $M \models \varphi$ is usually defined, we know implicitly that for some $L$, $M$ is an $L$-structure and $\varphi$ is an $L$-sentence.
I'm curious what the downsides are to defining $M \...
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Making sense of: "there is an uncountable set" is a logical consequence of $\mathrm{ZFC}$, yet $\mathrm{ZFC}$ is satisfiable in a countable domain.
As I learn the Compactness and Löwenheim-Skolem theorems of first-order logic and I begin to have a better understanding of what they really mean, something has me baffled.
As I have learned it, the ...
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Projecting Skolem's Paradox Upwards
My understanding of the resolution of Skolem's Paradox is that although in a countable model of ZFC there does not exist a bijection between a countable set and its powerset, we can still construct a ...
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elementary embeddings and ultrafilters
In Kanamori's book on large cardinals (second edition), on page 300, he is proving 22.4 Proposition (d), where the proposition says that if $U$ is an $\omega_1$-complete ultrafilter over a set $S$, ...
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Witnesses in the Henkin construction and the canonical model
See the images below, from Model theory: An Introduction, by David Marker: In the proof of Lemma 2.1.7 the canonical model is defined using equivalence classes of constant symbols of $\mathcal{L}$, ...
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Link between Tarski Truth, Negation, Consistency and Deductions
Based on the comments I have added an omission to formula (1) which does affect the question I asked, which is hopefully now more specific.
In Enderton "A Mathematical Introduction to Logic" ...
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Is this type of "definable bijective solution" of combinatorial identities symmetric?
Say that a combinatorial problem is a pair of first-order sentences $\langle\varphi,\psi\rangle$ in (disjoint and relational, for simplicity) finite languages $\Sigma=\{R_1,...,R_l\},\Pi=\{S_1,...,S_m\...
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Model of $ZFC$ where the internal powerset is the external powerset
Is there a name for models $M$ of $ZFC$ where the internal powerset is the external powerset?
If so, is the assumption of a transitive model where the internal powerset is the external powerset a ...
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Is there such a thing as the second-order theory of a structure?
Given a structure, say for example, $(\mathbb{R};+,*,0,1,<)$, I know the definition of the first-order theory of that structure. But is there such a thing as the second-order theory of a structure, ...
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Are the additional constant-like things in an elementary diagram parameters or are they fresh but otherwise unconstrained constant symbols?
Are the additional constant-like things in an elementary diagram parameters or are they fresh but otherwise unconstrained constant symbols?
For context, I've seen the theorems saying that we have a ...
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Why does a transitive class that is closed under Gödel operations and is almost universal satisfy Separation?
In Jech's Set Theory, Theorem 13.9 states the following.
A transitive class $M$ is an inner model of ZF if and only if it is closed under Gödel operations and is almost universal, i.e., every subset $...
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Denotation - Logic
I am currently studying the proof of the undecidability of dyadic logic from the book Computability and Logic (see section 21.3), written by George S. Boolos and John P. Burgess. I came across the ...
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When do definable functions have definable inverses?
Are there any conditions on a structure so that every definable function also has a definable inverse?
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Is a type-definable set (with parameters) invariant under arbitrary automorphism $\varnothing$-type-definable? [duplicate]
This statement is true for definable sets, but how about type-definable sets?
I encountered with this question when reading the proof of existence of $G^{00}$ for $\varnothing$-type-definable group $G$...
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Which boolean algebras are isomorphic to the algebra of definable sets for some structure with specific parameters?
Every boolean algebra is the algebra of clopen sets of some topological space, by Stone's representation theorem.
Not every boolean algebra is isomorphic to the powerset algebra of some set, though. ...
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