# 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 ...
1 vote
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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 $\neg \neg A$ ...
1 vote
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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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### 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 $s^M(\overline{a}) = a_{ij}$ ...
1 vote
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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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### 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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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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### 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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