# 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.

2,345 questions
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### About a topological proof of the compactness theorem

I'm trying to prove compactness theorem following this paper https://www.staff.science.uu.nl/~ooste110/syllabi/eric-poizat.pdf. Let $\mathcal{T}$ be the set of all complete theories over a fixed ...
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### if $F\subseteq K$ real closed fields, $b\in K$ algebraic over F then $b\in F$

I'm studying real closed field part in model theory. Because I'm not familiar with abstract algebra, I'm stuck with two (possibly) simple problems. Here are definitions and two problems I'm stuck with....
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### Axiomatizing a “bounded” companion to PA

There's nothing special about PA here, I'm just focusing on it since it's strong enough to ignore lots of minor technical issues around foundations. If switching to some other theory would yield a ...
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### How is mathematics formulated - with models of formal systems?

How does one start doing 'Mathematics' from the ground up? My naive intuition (I don't know any logic) was that it's something like this: 1) Write down some symbols; so that we can use them for ...
Consider the $1$-types over $\mathbb{Q}$ as a dense linear order without endpoints. They are similar to cuts: for any $1$-type $p$ over $\mathbb{Q}$, we can define $L_p = \{a \in \mathbb{Q} \; | \; a &... 1answer 26 views ### If$T$admits quantifier elimination in$\mathcal{L}$, does it admit quantifier elimination in$\mathcal{L}(c)$? I know this is true: If$T$is an$\mathcal{L}$-theory and it admits quantifier elimination in$\mathcal{L}(c)=\mathcal{L}\cup\{c\}$, where$c$is a constant symbol not in$\mathcal{L}$, then$T$... 1answer 33 views ### A necessary and sufficient condition for a structure to be rigid. Is this a necessary and sufficient condition for a structure$M$to be rigid: For all distinct$m, m'$in the underlying set of$M$,$Th(M,m) \neq Th(M,m')$? 0answers 34 views ### Satisfication of modal formulas in ultrapower modals I am learning from Blackburn's modal logic book, and I am attempting proving: Proposition 2.71 Let$\prod_U M$be an ultrapower of$M$. Then, for all modal formulas$\phi$we have$M,w\vdash \phi$... 1answer 55 views ### Show that the set of real numbers is not definable in the field of complex number Like what the title suggest, I wish to show that$\mathbb{R}$cannot be define in$\mathbb{C}$. I want to make use of the following proposition ; (David Marker) Fix a structure$M$, if$X \subset M$... 2answers 230 views ### Axiom checking as type checking? There is a connection between type theory and logic, where types are propositions, and type checking performs the role of checking whether a proof of a proposition is correct (Curry-Howard isomorphism)... 1answer 115 views ### Vacuous truths in Superstructure approach to Nonstandard Analysis Good evening everybody, at the moment I'm studying non-standard analysis, specifically the superstructure approach to it. This approximately works as described in chapter 3 of http://people.dm.unipi.... 1answer 33 views ### Is the structure$(\mathbb{R}, +, *)$rigid? [duplicate] Does the structure$(\mathbb{R}, + ,*)$have only the trivial automorphism? 1answer 27 views ### A not-$\omega$-saturated model. I'm new to$\omega$-saturated model and the likewise and although I'm aware of examples of$\omega$-saturated models$(\mathbb{Q},<)$, I can not really imagine a not-$\omega$-saturated model and ... 1answer 166 views ### Complete Theorem Let$\mathcal{L}$be a language with the constants${a_1},{a_2}$and the single parameter operation$F$. Looking at the set$\Gamma=\{ \psi,\chi,\eta \}\cup\{\phi_n|n\ge1\}$where$\psi=\forall{x}(x\...
It is well known, as Fraïssé's theorem, that for a finite relational signature $\sigma$ and two $\sigma$-models $\mathfrak{A}$, $\mathfrak{B}$, $\mathfrak{A}$ and $\mathfrak{B}$ are elementary ...