# 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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### Model existence for infinitary logics

One of the problems of infinitary logic is that it is possible for compactness to fail in a spectacular way: for example, one can concoct an inconsistent set of axioms whose proper subsets are all ...
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### Why can't we prove consistency of ZFC like we can for PA?

this might be a silly question, but I was wondering: PA cannot prove its consistency by the incompleteness theorems, but we can "step outside" and exhibit a model of it, namely $\mathbb{N}$, so we ...
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### What kind of compactness does "expanding $\mathbb{R}$ by constants" have?

EDIT: Now crossposted at mathoverflow. This arose from my answer to another question. Say that a theory $T$ in the language of ordered fields + constants is $\mathbb{R}$-satisfiable if it has a model ...
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### Expressing associativity with only two variables

I'm wondering if it is possible to axiomatize associativity using a set of equations in only two variables. Suppose we have a signature consisting of one binary operation $\cdot$. Is it possible to ...
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### Comparing countable models of ZFC

Let us consider the class $\cal C$ of countable models of ZFC. For ${\mathfrak A}=(A,{\in}_A)$ and ${\mathfrak B}=(B,{\in}_B)$ in $\cal C$ I say that ${\mathfrak A}<{\mathfrak B}$ iff there is a ...
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### In what sense of "structure" do group homomorphisms "preserve structure"?

It is commonly said that group homomorphisms "preserve the structure of the group", e.g., from Wikipedia: The purpose of defining a group homomorphism as it is, is to create functions that preserve ...
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### Showing any countable, dense, linear ordering is isomorphic to a subset of $\mathbb{Q}$

I'm trying to knock out a few of the later exercises from Enderton's Elements of Set Theory. This problem is #17, found on page 227. A partial ordering $R$ is said to be dense iff whenever $xRz$, ...
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### What is Model Theory

I've been reading up on some Model Theory lately. My question is simple. What is the purpose of the theory to begin with? How does it enrich mathematics? A-priori, it doesn't seem like we're doing ...
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### Has a conjecture ever originally been decided by constructing the proof with mathematical logic?

So, one of the things that mathematical logic does is study theorems as abstract objects. There also many theorems about mathematical logic, and these theorems can have connections to other fields. ...
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### What is the definition of a definition?

In mathematical logic or other formal systems, what is the definition of a definition, formally? If "A" is defined as "B", what is the definition of "A" like? Does it ...
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### Non-axiomatisability and ultraproducts

Let $T$ be a first-order theory over a language $L$, and let $\mathcal{M}$ be a subclass of the class of models of $T$. As I understand it, if there is no theory $\hat{T}$ over $L$ whose class of ...
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### Does a finite first-order theory which has a model always have a finite model?

I'm curious whether this is true or not: Let T be a finite, first-order theory. If T has a model, then T has a finite model. I would assume the answer is 'yes', but I wanted to make sure I haven't ...
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### How can we know we're not accidentally talking about non-standard integers?

This question is mostly from pure curiosity. We know that any formal system cannot completely pin down the natural numbers. So regardless of whether we're reasoning in PA or ZFC or something else, ...
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### Most astonishing applications of compactness theorem outside logic

The compactness theorem has a lot of applications to logic and model theory. I'm looking for applications. I'm looking for theorems in other areas of mathematics which seem at first sight to have ...
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### Can Russell's paradox, Halting problem and Godel's Incompleteness theorem be generalized?

All these three theorems (I am not 100% sure about the third, but I have heard it has a similar argument with the other two) use self-referentiability as contradiction and they talk about the ...
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### Are there non-standard counterexamples to the Fermat Last Theorem?

This is another way to ask if Wiles's proof can be converted into a "purely number-theoretic" one. If there is no proof in Peano Arithmetic then there should be non-standard integers that satisfy the ...
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