# Tagged Questions

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 ...

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### Can there be a countable transitive model satisfying the same $MK$ theory as $V$?

A little while ago, I asked whether or not there could be a countable transitive model satisfying the same $ZFC$ theory as $V$ (assuming that we're working within some $V$, or (if you like) that there ...
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### Defining a formula using FO+TC

Define a signature Σ and an FO+TC formula ϕ over Σ, such that: there is no infinite structure satisfying ϕ for every even natural number n>0 there is a structure of size n satisfying ϕ for every ...
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### model theory & algebraically closed field

recently I'm taking mathematical logic course, and the class covers some basic model theoretical ideas. Since I have not taken any abstract algebra course, It is so hard to understand what is going on....
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### How should one read the s*(t) function in Mendelson's Introduction to Mathematical Logic?

I'm self-teaching logic and doing it by means of following Elliot Mendelson's Introduction to Mathematical Logic (6th edition). In p.56 he defines a a function s* which, in his words, 'assigns to each ...
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### Countable elementary submodels

I'm having some trouble understanding elementary submodels. Let $H_\chi$ be the set of all sets which are hereditarily of cardinality $<\chi$. Let $\textbf{N}=(N,\in)$ be a countable elementary ...
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### Why in countably saturated models, types that are consistent with $TH(\mathcal{M_a})$ are finitely realizable?

I'm learning about countably saturated ($\alpha$-saturated) models. There is a hidden presupposition everywhere used: The type $\Gamma(x)$ is consistent with $TH(\mathcal{M_a})$ iff $\Gamma(x)$ is ...
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### Is model theory needed to understand ordinal logic?

By ordinal logic, I mean turing's ordinal logic. I'm going to learn first order logic, elementary set theory, basic computability, and godelian incompleteness as prerequisites for ordinal logic. But, ...
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### Let $L$ be a first order language with equality and two binary function symbols…

$\bf Question$ Let $L=(F=\{f^2,g^2\}, P=\{=\}, C=\{c\})$ be a first order language. Let $I = (\Bbb R^{2\times 2}, f^2(x,y)=(x+y)^t,g^2(x,y)=x+y,0)$ be an interpretation of $L$. Exhibit ...
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### Can't EF game theory be applied to finite languages WITH function symbols?

Let $\mathcal{M}$ and $\mathcal{N}$ be two structures in a language $\mathcal{L}$. We define the finite determined game $G_n(\mathcal{M},\mathcal{N})$ as a game with $n$ rounds where in each round ...
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### non-isomorphic countable models of $Th(\mathbb{N})$

I'm proving there are exactly $2^\omega$ non-isomorphic countable models of standard natural numbers. I got cardinality of them $\geq 2^\omega$from prime arguments. but I don't get how to prove other ...
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### Is this an axiomatization of real closed fields?

I know that real closed fields are defined as ordered fields where every positive element is a square and every odd polynomial has a root. But can they also be axiomatized as totally ordered fields ...
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### Is logical implication examining only the syntatical component?

As my title asks, is logical implication only examining the syntatical compinent of the formal language? I am using Enderton's book on Mathematical Logic in my class and after some work I am ...
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### Show the inequality $x <y$ is definable in the language $\langle \mathbb{R}; +, \times ; 0,1 \rangle$

My initial idea is that I need to find a sentence that expresses 'x is positive' and then I can say: for any $a, b$, $a>b$ iff there is a positive x s.t. $b+x=a$, but can't figure out how, any ...
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### A simple proof that elementary equivalence and isomorphism coincide for finite structures?

I'm wondering if there's a straightforward proof of this result I've seen mentioned in quite a few places. If $\mathcal{L}$ is finite, of course, this is trivial since there's a single formula that ...
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### Is a theory elementary iff it is categorical? [closed]

If an elementary class is the set of structures satisfying a theory, and a theory is categorical if it determines a structure up to an isomorphism, it would seem that the two concepts are related, no?
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### bi-interpretability and automorphism groups

Let $M$ and $N$ be two first order structures, say they are countable and $\aleph_0$-categorical. Then $M$ and $N$ are bi-interpretable if and only if their automorphism groups $Aut(M)$ and $Aut(N)$ ...
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### Show the only definable subsets of $\langle \mathbb{Q} , < \rangle$are the empty set and $\mathbb{Q}$

I'm trying to show the above, and I know it should be quite simple, but I'm struggling to get my head around how you show subsets are not definable using automorphisms- so a detailed explanation would ...
Let $F$ be a real closed field. It is known$^{[1]}$ that $F$ has an integer part, that is, a subring $A$ such that $\forall x \in F, \exists ! a \in A, a \leq x < a+1$. Are all integer parts over ...