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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Nonsingular zeroes are algebraic?

I'm getting started in Real Algebraic Geometry (from a model-theory perspective), and a paper makes the following assertion (here $K\subset L$ are real-closed fields): Suppose $Q\in L^n$, $f_1, ...
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305 views

How do we avoid circularity when we build a structure for ZFC? [closed]

when investigating ZFC as a formal language a structure is a set, are we not engaging in circular logic here? Or is 'set' thought of in a more primitive sense?
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626 views

Intuition behind o-minimal structures.

This is very much the same post as I posted at math.stackexchange. I am following the definitions of an o-minimal system in "Tame Topology and O-minimal Structures" by Lou van den Dries. It is ...
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165 views

Characterizing $\mathbb{Q}$ among number fields

Is there an $\mathcal{L}_{\omega_1\omega}$ formula or set of formulas that characterizes the rationals $\mathbb{Q}$ among other number fields? EDIT: My formula must not contain an infinite number of ...
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146 views

Is this fragment of arithmetic on $p^{-\infty} {\mathbb Z}$ decidable?

Let $p$ be a prime number. Consider the abelian group $p^{-\infty} {\mathbb Z} = \bigcup p^{-n} {\mathbb Z}$ consisting of rational numbers whose denominator is a power of $p$, under addition. View ...
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338 views

Interpretations as morphisms

Model-theoretic interpretations seem to give rise to categories in which morphisms are not functions. To name just two small examples: the category of finite graphs with interpretations between them ...
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789 views

Where do nonstandard elliptic curve angles come from?

This is a question which has bounced around my head over the past few years. At the same time, I am answering ...
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310 views

generalizing the ultrapower

Given an 2-valued measure $\mu$ on a set $I$, and structures $M_i \ (i \in I)$, one can construct the ultrapower $\prod_{i\in I}M_i / U$ (where $U$ is the ultrafilter associated with $\mu$.) One can ...
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327 views

Riemann hypothesis for zeta function of definable sets over finite fields

Hi, Consider the zeta function of a definable set over a finite field. More precisely, let $\varphi$ be a formula in the language of fields and let $X$ be a definable subset of $F^n$ given by ...
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72 views

local character of Tarski-Vaught for tuples in excellent classes

In the book Baldwin, Categoricity in Abstract Elementary Classes defines (Def.20.1,p.151) a notion of Tarski-Vaught extensions for tuples that generalises both independence and usual Tarski-Vaught ...
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3answers
299 views

computing Morley rank using parameters from an arbitrary model

One of the ways to define the Morley rank of a definable set is with respect to a model, say $M$, i.e. a set has rank $\alpha+1$ if there are infinitely many definable subsets with parameters in $M$ ...
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768 views

Axioms for Riemann $\zeta$ function

Are there any set of axioms that completely characterize the Riemann zeta function? i.e. like Ressayre axioms for the exponential function in an exponential field or functional equations.
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405 views

Cantor theorem on orders

It is "a well-known theorem of Cantor", said Sierpinski (circa 1920), that every countable total order can be imbedded in the rationals, and he proceeds to demonstrate that, assuming the continuum ...
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202 views

an example of a strictly superstable field

It is well-known that the theory of separably closed fields of some fixed positive characteristic and degree of imperfection is stable but not superstable. By a result of Cherlin and Shelah a ...
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2answers
321 views

The category of Boolean-valued models associated to a model of ZFC

This is related to my previous question here, and indirectly motivated by Andreas Blass intriguing answer therein: start from a ZF transitive model $M$, and consider the category $CBA(M)$ of ...
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2answers
598 views

All properties of a mathematical object

This is primarily a question about related literature. I am looking for specific references, or terminology that I can use to search for references. Let A a well defined mathematical structure of ...
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1answer
139 views

Feferman-Kreisel preservation theorem

I want to show the following theorem from Feferman and Kreisel: Let $\phi$ be such that there is a theorem $\sigma$ of ZFC so that for every transitive model $M$ of $\sigma$, we have: $\phi^M \to ...
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140 views

axiomatizing the abelian part of the topological fundamental groupoid functor on algebraic varieties

let Vv be the category of complex algebraic varieties defined over $\bar Q\subset C$, and let $\pi_1^{top}:Vv\longrightarrow Groupoids$ sending a variety V into its (strict) fundamental groupoid ...
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171 views

Is there a general theory of models that has as instances classical FOL, classical propositional logic, etc.?

Is there any general theory of models that has as instances classical FOL, classical propositional logic, etc.?
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630 views

Elementary Equivalence =? Homotopy Equivalence

One of the most interesting novelties in recent foundational studies is Voevodsky's Homotopical Type Theory project (see here). Finally homotopy theory ideas have entered in a royal fashion the ...
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255 views

Can every $\mathcal{L}_{\omega_1,\omega}$ formula be expressed as a type? What about canonical forms?

If $\mathcal{L}$ is a countable, first-order language, it is easy to see that every $n$-type $p$ (over $\emptyset$) can be expressed as an $\mathcal{L}_{\omega_1,\omega}$-formula, namely ...
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217 views

Uncountable Lüroth problem

Question. Let $F(X)$ be the field obtained by adding an uncountable collection of indeterminates (mutually transcendental elements) to a prime field $F$. Is there an example of a subfield $E$ ...
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413 views

How badly does compactness fail in $\mathcal{L}_{\omega_1\omega}$?

I would like to get a better idea of how badly compactness fails in $\mathcal{L}_{\omega_1\omega}$. Let $\Gamma$ be an arbitrary set of sentences from $\mathcal{L}_{\omega_1\omega}$. Let the ...
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316 views

Is there a statement equivalent to a sentence admitting $(\alpha^{+n},\alpha)$?

From Chang and Keisler's "Model Theory", section 7.2, we know that: 1) There is a sentence $\sigma$ in a suitable language $L$ such that for all infinite cardinals $\alpha$, $\sigma$ admits ...
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188 views

Open question in abstract model theory

The existence of an extension of first order logic satisfying both the Compactness Theorem and the Interpolation Theorem is an open or solved question?
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Are there any complete, first-order and unstable theories which have non-categorical second-order formulations?

Since it's not stable, $PA$ fails at being categorical in a power in the worst possible way, having $2^{\lambda}$ models in any uncountable $\lambda$. But $PA$ regains its categoricity in the move to ...
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Is there exponentiation in “sufficiently large” models of $I\Delta_{0}$?

Let $L_{E}$ be the language of discretely ordered rings together with an extra predicate symbol $E$. The system $A$ consists of the axioms of $I\Delta_{0}$ (basic arithmetic plus induction for bounded ...
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201 views

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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Terminology for system of equations and…

I am looking for the standard term for a system that consists of things of the form $p_i(x_1,\ldots ,x_n)=0$ and of the form $q_j(x_1,\ldots,x_n)\neq 0$ with the $p_i$ and $q_j$ polynomials. I have ...
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472 views

Countable admissible ordinals

Jensen claimed that for any finite increasing sequence countable admissible ordinals $\omega= \alpha_0<\alpha_1\cdots <\alpha_n$, there is a real $x$ so that, for each $m\leq n$, $\alpha_m$ is ...
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606 views

What is the etymology of model?

What is the etymology of model? The answer is of course pre-WWW, but the better part of an hour in the library searching both classic model theory and modal logic textbooks turned up nothing. Every ...
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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 ...
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Is ramification of number fields first order?

Fix a prime number $p$. Is there a first order sentence $\phi_p$ in the language of fields such that $\phi_p$ holds in a number field $K$ if and only if the prime $p$ is unramified in the field ...
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Elementary end extension of a countable model for ZF

Theorem 2.2.18 in Chang and Kiesler uses omitting types to show that any countable model of ZF has an elementary end extension. Can we control the countable order type of such a model? for example, if ...
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540 views

Reference Request: Non-Standard Models of PA

I am attempting to write an expository paper on non-standard models of PA that is accesible to students taking an introductory graduate course in mathematical logic (covering Godel's incompleteness ...
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513 views

Measure of progress towards a proof

Can one define some measure of progress towards a proof of a statement? I'm not sure if it's even possible for general first order logic statements so let's restrict ourselves to propositional ...
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964 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 ...
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Proofs in the same vein as Ax-Grothendieck

I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...
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Formal proof of Con(ZFC) => Con(ZFC + not CH) in ZFC

Is it possible to prove $Con(ZFC) \rightarrow Con(ZFC + \neg CH)$ purely within ZFC? To prove this (using forcing) one seems to need a countable transitive model of ZFC. The texts I am reading avoid ...
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197 views

Defining $\mathbb{Z}$ in $\mathbb{Z}^\ast/P\mathbb{Z}^\ast$ where $P$ is an infinite prime

In continuation of my recent questions, here is the last one: Is there a simple formula preferably existential that defines $\mathbb{Z}$ in $\mathbb{Z}^\ast/P\mathbb{Z}^\ast$ where $\mathbb{Z}^\ast$ ...
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1answer
209 views

Defining $\mathbb{Z}$ in $\prod_p \mathbb{F}_p(t)/\mathcal{U}$

Is it possible to show that there is a simple formula, preferably existential, that characterizes a nonstandard model of the ring of integers among the elements of $\prod_p ...
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1answer
332 views

Defining $\mathbb{Z}^*$ in $\prod_p \mathbb{F}_p/\mathcal{U}$ (or pseudo-finite fields)

Is it possible to show that there is a simple formula, preferably existential, that characterizes a nonstandard model of the ring of integers among the elements of $\prod_p \mathbb{F}_p/\mathcal{U}$? ...
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404 views

Ultraproduct $\prod_p \mathbb{F}_p/\sim$ and $\mathbb{Q}^*$

Is the non-principal ultraproduct of finite fields $\prod_p \mathbb{F}_p/\sim$ a nonstandard model of the rationals $\mathbb{Q}$? EDIT: Can we realize $\mathbb{Q}^*$ as an ultraproduct?
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775 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 ...
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Boolean-Valued Models vs. the Infinite-valued Logic of Lukasiewicz and set theory

Is anyone familiar with an old paper of C.C. Chang entitled "The Axiom of Comprehension in Infinite-Valued Logic" which shows that the Axiom of Comprehension without parameters is consistent in the ...
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906 views

Nonstandard Methods ( or Model Theory ) and Arithmetic Geometry

I hear that the nonstandard methods are applied to many problems in various fields of mathematics such as functional analysis, topology, probability theory and so on. So, I have become interested in ...
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On Grothendieck ring and semiring

We are given a language $L$ and a structure $M$ (model). Definable sets in this model are subsets of $M^n$ definable by a formula of $L$. The Grothendieck semiring of the structure is defined in the ...
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874 views

Is there an analogue of finite fields for products of two prime powers?

The collection of prime powers can be characterized in the following way: There is a field with $q$ elements if and only if $q$ is a prime power. Furthermore if it exists then this field is unique ...
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679 views

Is there a model of Set Theory which thinks it is the Standard Model, i.e. is there a Universe U such that U $\models$ U=V?

I asked my friend (a Set Theorist) this question and he said that every model of ZFC thinks it is the Standard Model. But, I'm not sure it is so simple. First, because I don't know how a Universe ...
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307 views

Does model-complete in a language with a constant symbol imply EQ?

Marker Theorem 3.1.4 says: Suppose $T$ is a theory in a language with at least one constant symbol. Then an $L$-formula $\phi(x)$ is $T$-equivalent to a quantifier-free formula iff, whenever $M$ and ...