# Tagged Questions

Model theory is the branch of mathematical logic which deals with the connection between a formal language and its interpretations, or models.

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### Is there a 0-1 law for the theory of groups?

Several months ago, Dominik asked the question Is there a 0-1 law for the theory of groups? on mathstackexchange, but although his question received attention there is still no answer. By asking the ...
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### What are some results in mathematics that have snappy proofs using model theory?

I am preparing to teach a short course on "applied model theory" at UGA this summer. To draw people in, I am looking to create a BIG LIST of results in mathematics that have nice proofs using model ...
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### Is the ultraproduct concept fundamentally category-theoretic?

Once again, I would like to take advantage of the large number of knowledgable category theorists on this site for a question I have about category-theoretic aspects of a fundamental logic concept. ...
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### Has decidability got something to do with primes?

Note: I have modified the question to make it clearer and more relevant. That makes some of references to the old version no longer hold. I hope the victims won't be furious over this. Motivation: ...
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### Heuristic argument that finite simple groups _ought_ to be “classifiable”?

Obviously there exists a list of the finite simple groups, but why should it be a nice list, one that you can write down? Solomon's AMS article goes some way toward a historical / technical ...
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### Is $π$ definable in $(\Bbb R,0,1,+,×,<,\exp)$?

(This question is originally from Math.SE, where it didn't receive any answers.) Is there a first-order formula $\phi(x)$ with exactly one free variable $x$ in the language of ordered fields ...
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### Is the field of constructible numbers known to be decidable?

By the field of constructible numbers I mean the union of all finite towers of real quadratic extensions beginning with $\mathbb{Q}$. By decidable I mean the set of first order truths in this field, ...
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### What's a magical theorem in logic?

Some theorems are magical: their hypotheses are easy to meet, and when invoked (as lemmas) in the midst of an otherwise routine proof, they deliver the desired conclusion more or less straightaway&...
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### Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic

Mathematician Edward Nelson is known for his extreme views on the foundations of mathematics, variously described as "ultrafintism" or "strict finitism" (Nelson's preferred term), which came into the ...
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### Where do uncountable models collapse to?

Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ...
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### Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,…)

Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ... On the other hand model theory, in particular after Hrushovski, found many ...
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### Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
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### Isomorphism types or structure theory for nonstandard analysis

My question is about nonstandard analysis, and the diverse possibilities for the choice of the nonstandard model R*. Although one hears talk of the nonstandard reals R*, there are of course many non-...
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### Which graphs are elementarily equivalent to their own disjoint sums?

In Stefan Geschke's recent question, one of the solutions observed that the graph consisting of a single infinite beaded chain, a $\mathbb{Z}$-chain where each integer is connected to its nearest ...
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### Completeness vs Compactness in logic

One standard approach to showing compactness of first-order logic is to show completeness, of which compactness is an easy corollary. I am told this approach is deprecated nowadays, as Compactness is ...
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### What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a model of the hyperreals? Or in a nonstandard model of arithmetic?

I am interested to know the full range of possibilities for the cofinality type of cuts in an ordered field and in other structures, such as nonstandard models of arithmetic. Definitions. ...
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### Is non-existence of the hyperreals consistent with ZF?

I know that it is possible to construct the hyperreal number system in ZFC by using the axiom of choice to obtain a non-principal ultrafilter. Would the non-existence of a set of hyperreals be ...
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### Universe view vs. Multiverse view of Set Theory

Here I refer to Hamkins' slides: http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf particularly, to the "Universe view simulated inside Multiverse", p. 22. My question is: is it very unsound ...
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### Axiomatizations of the real exponential field

According to Marker's "Model Theory: An Introduction", the real exponential field has a $\forall\exists$ axiomatization (because it is model complete) but no-one has any idea what such an ...
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### Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s). Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following: (a) Trivial (...
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### Application of Fraïssé construction in set theory

As you know Fraïssé limit construction and its generalization, Hrushovski's construction, have many applications in model theory to build models with interesting property. Now I would like to know ...
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### In model theory, does compactness easily imply completeness?

Recall the two following fundamental theorems of mathematical logic: Completeness Theorem: A theory T is syntactically consistent -- i.e., for no statement P can the statement "P and (not P)" be ...
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Yesterday, in the short course on model theory I am currently teaching, I gave the following nice application of downward Lowenheim-Skolem which I found in W. Hodges A Shorter Model Theory: Thm: Let $... 5answers 2k views ### Category theory and model theory as “natural” counterparts I am aware of the profound discussion of the relationship between category theory and model theory (e.g. at The n-Category Café) but do wonder why category theory (CT) is not opposed to model theory (... 2answers 2k views ### Is there a known natural model of Peano Arithmetic where Goodstein's theorem fails? (I've previously asked this question on the sister site here, but got no responses). Goodstein's Theorem is the statement that every Goodstein sequence eventually hits 0. It turns out that this ... 4answers 1k views ### Non-ZFC set theory and nonuniqueness of the hyperreals: problem solved? The reals are the unique complete ordered field. The hyperreals$\mathbb{R}^\*$are not unique in ZFC, and many people seemed to think this was a serious objection to them. Abraham Robinson responded ... 0answers 924 views ### Defining$\mathbb{Z}$in$\mathbb{Q}$It was proved by Poonen that$\mathbb{Z}$is definable in$\mathbb{Q}$using$\forall \exists$formula. Koenigsmann has shown that$\mathbb{Z}$is in fact definable by universal formula. What is the ... 2answers 979 views ### nonstandard models and mathematical theorems Is there a first order statement about the natural numbers (not nonstandard analysis) such that the truth of the statement is easier to see in a nonstandard model? In other words, do nonstandard ... 3answers 929 views ### Can we recognize when a category is equivalent to the category of models of a first order theory? Many of the most canonical early examples of categories arise as the collection of models of a fixed first order theory, with the related model-theoretic concept of homomorphism. For example, the ... 1answer 577 views ### Is the class of additive groups of rings axiomatizable? I know that it is impossible to axiomatize the multiplicative structures of rings, called$R$-semigroups. Is anything known about the first-order axiomatizability of the class of abelian groups which ... 1answer 216 views ### Existence property for ordered fields A theory$T$has the existence property (EP) if the following holds: Let$\phi(x)$be a formula with one free variable (and no parameters) such that$T \vdash (\exists x) \phi(x)$. Then there is ... 0answers 374 views ### What is the Cantor-Bendixson rank of the space of first order theories? Let$L$be the language$\{R\}$containing a single binary relation symbol. Consider the space$S_0(L)$of complete, first-order$L$-theories. This is a seperable, compact Hausdorff space; what is its ... 2answers 968 views ### When does Vopěnka's principle hold? Vopěnka's principle (VP) is the statement that, given any proper class$\{\mathcal{A}_\eta: \eta\in ON\}$of first-order structures in the same language, there are some$\alpha\not=\beta$with$\...
I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way? Let $\mathcal{C}$ be a class of (...
In this topic, I will use the word uncountable group referring to groups whose cardinality is $\leq|\mathbb R|$. Notation: $R$ is the hyperfinite $II_1$-factor, $\omega$ is a free ultrafilter on the ...