# 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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### Independence in mathematics

While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over ...
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### the structure on the value group sort of a C-minimal field

Let $K$ be an algebraically closed valued field which is $C$-minimal, as defined, for example, in this article. Examples include pure algebraically closed valued fields, as well as Lipschitz and ...
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### Why do we need a transitive model in forcing arguments?

One major approach to the theory of forcing is to assume that ZFC has a countable transitive model $M \in V$ (where $V$ is the "real" universe). In this approach, one takes a poset $\mathbb{P} \in M$, ...
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### Consistent sentences with no arithmetically definable models

I've seen a construction of a sentence of first order logic that is consistent, but has no models with underlying set $\mathbb{N}$ and recursive functions and relations. Do there also exist consistent ...
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### Constructive Resolution of Toric Singularities via Model Theory

Do there exists some language $\mathcal{L}$ of rational polyhedral cones in rational vector spaces and a theory $T$ over $\mathcal{L}$ whose models $\mathcal{M}$ are resolutions of toric singularities?...
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### A kind of saturation property related to forcing notions

Forcing is typically done over well-founded models. There are lots of good reasons for this, but it can feel confining at times. Fortunately, we can equally well force over non-well-founded models! It ...
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### Construction of model of arithmetic from an arbitrary model

Let $M$ be a non-standard model of $PA$, $a\in |M|$ be an arbitrary non-standard number and $T$ be a theory of arithmetic. We want to choose a subset $M'\subsetneq M$ such that: $M'\models PA^-$ (or ...
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### O-minimal spectrum is a spectral space

I'm trying to understand a proof on "Sheaves of Continuous Definable Functions" (Pillay, Anand. "Sheaves of continuous definable functions." The Journal of symbolic logic 53.04 (1988): 1165-1169.) ...
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### Model existence and consistency conditions for $\Pi_1^0$ oracles

Let a $\Pi_1^0$ sentence be a sentence asserting that some given Turing machine never halts at the empty input tape. Let Q1 be a (potentially consistently lying) oracle for deciding $\Pi_1^0$ ...
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### Is there a model of ZF+ACC where transfer fails for the definable hyperreals?

A decade ago Kanovei and Shelah constructed a definable hyperreal field. The ultrapower used exploits a fairly large index set so that it is clear that the usual proof of Los and transfer does not go ...
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### Example of $\aleph_1$-categorical linear order

Is it possible to have an $L_{\omega_1,\omega}$-sentence $\phi$ in a vocabulary that includes $<$ that satisfies the following? $<$ is a linear order on a definable subset; $\phi$ is $\aleph_1$...
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### On the theory of infinite extraspecial $p$-groups

$p$ is a prime number. A group $G$ is called an infinite extraspecial $p$-group if 1) it is infinite, 2) every $g\neq 1$ in $G$ has order $p$, 3) its centre $Z(G)$ coincide with $G'$ and is a ...
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### When are generic models not too wild?

This is a question related to ideas raised in http://arxiv.org/abs/1410.1224 and http://arxiv.org/pdf/1405.7456.pdf. Basically, the idea is the following: Suppose I have a first-order theory $T$. ...
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### Are Braid Groups with Finitely many Generators NIP?

I am curious what braid groups (strings in $\mathbb{R}^3$) are NIP. Consider the following: Let $B_\mathbb{N}$ be braid group with "braids" indexed by the natural numbers (alternatively, the direct ...
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### Non-Archimedean non-standard models for R

Let $\langle \mathbb{R}, 0, 1, +, \cdot, <\rangle$ be the standard model for $R$, and let $S$ be a countable model of $R$ (satisfying all true first-order statements in $R$). Is it true that the ...
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### What is the precise relationship between o-minimal theory and Grothendieck's “Esquisse d'un programme”?

I have seen various references in the literature to such a connection but they tend to assume that the reader is familiar with the connection, and limit themselves to providing additional detail. So ...
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### Predicates of infinite arity

Infinitary logic considers languages being infinite by infinite conjunctions and disjunctions. I wonder why it not considers languages being infinite by relations and functions of infinite arity. ...
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### Current Main Areas of Research in Model Theory [closed]

Could someone gives a general picture of the present state of Model Theory as a field? What are the current main areas and directions of research? What are some examples of the current experts and the ...
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### Deciding isomorphism between structures which interpret in the pure set

I am interested in the following decision problem: Given descriptions of two relational structures $A,B$ which interpret in the pure set $\mathbb N=(\{0,1,2,\ldots\},=)$, decide whether $A$ and $B$...
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### Variant of Conceptual Completeness

Let $\mathcal{C}$ and $\mathcal{D}$ be pretopoi, and let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a pretopos functor (that is, a functor which preserves finite coproducts, finite limits, and ...
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### What are the advantages of the more abstract approaches to nonstandard analysis?

This question does not concern the comparative merits of standard (SA) and nonstandard (NSA) analysis but rather a comparison of different approaches to NSA. What are the concrete advantages of the ...
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### What is the modern consensus on the difficulty of infinitesimals?

At a related thread at MSE an expert in reverse mathematics noted that "As the modern consensus is that only nonstandard models have infinitesimals, it will be quite challenging to give a concrete ...