**10**

votes

**1**answer

370 views

### Number of linear orders

It is well known that for every infinite cardinal $\kappa$ the number of non-isomorphic total orders of cardinality $\kappa$ is $2^\kappa$. Who first proved this, and in what context? Was it proved ...

**4**

votes

**3**answers

646 views

### Survey of finite axiomatizability for relational theories?

An $L$-theory $T$ is finitely axiomatizable if there is a finite set $A$ of $L$-sentences with the same consequences as $T$, i.e. such that $M \models T$ iff $M \models A$ for every $L$-structure $M$. ...

**3**

votes

**1**answer

202 views

### Nondegenerate involutions on the complex numbers — always just conjugation?

Suppose you have a ring homomorphism $(-)':\mathbb{C} \to \mathbb{C}$, which is an involution such that $\sum_i a_i a_i' = 0 \Leftrightarrow \forall i \ a_i=0$, where this sum is finite.
Must this ...

**2**

votes

**3**answers

358 views

### Methods for proving non FO definability

I'm having difficulties to prove that the subset of even numbers is not first-order definable in $(\mathbb{N},<)$. Any hint is welcomed!
More generally, what are usual techniques in order to prove ...

**-1**

votes

**1**answer

297 views

### A question on intuitionistic propositional logic [closed]

One week ago, I asked a question on math.stackexchange.com (http://math.stackexchange.com/questions/209120/a-question-on-intuitionistc-propositional-logic). But nobody answered my question. So I ...

**1**

vote

**1**answer

200 views

### Existentially closed substructure and ultraproducts

Is it true that if M is existentially closed in N then N can be embedded in an ultraproduct of M ?
Best regards.

**2**

votes

**2**answers

344 views

### Applications of the Ax Kochen Ershov (AKE) princicple

The AKE priciple states that two finitely ramified Henselian field (this means that the field is either of residual caracteristic 0 or is of characteristic $p$ and there is only a finite number of ...

**0**

votes

**1**answer

155 views

### 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, ...

**1**

vote

**2**answers

321 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?

**9**

votes

**2**answers

689 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 ...

**1**

vote

**0**answers

167 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 ...

**2**

votes

**0**answers

163 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 ...

**3**

votes

**1**answer

356 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
...

**6**

votes

**1**answer

792 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 ...

**5**

votes

**2**answers

411 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 ...

**1**

vote

**1**answer

333 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 ...

**4**

votes

**0**answers

75 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 ...

**6**

votes

**3**answers

339 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 $M$ ...

**8**

votes

**4**answers

820 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.

**3**

votes

**2**answers

445 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 ...

**4**

votes

**2**answers

225 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 ...

**3**

votes

**2**answers

344 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 ...

**2**

votes

**2**answers

613 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 ...

**2**

votes

**1**answer

143 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 ...

**3**

votes

**0**answers

149 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
...

**2**

votes

**3**answers

172 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.?

**11**

votes

**1**answer

698 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 ...

**4**

votes

**3**answers

284 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 ...

**8**

votes

**0**answers

240 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 $E$ ...

**11**

votes

**1**answer

439 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 ...

**4**

votes

**1**answer

341 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 ...

**1**

vote

**0**answers

196 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?

**5**

votes

**2**answers

467 views

### 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 ...

**6**

votes

**1**answer

269 views

### 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 ...

**0**

votes

**1**answer

211 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 ...

**2**

votes

**1**answer

153 views

### 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 ...

**5**

votes

**1**answer

546 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 ...

**10**

votes

**1**answer

648 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 ...

**7**

votes

**0**answers

346 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 ...

**11**

votes

**1**answer

372 views

### 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 ...

**5**

votes

**1**answer

345 views

### 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 ...

**7**

votes

**4**answers

577 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 ...

**3**

votes

**2**answers

519 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 ...

**13**

votes

**4**answers

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 ...

**9**

votes

**6**answers

1k views

### 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 ...

**6**

votes

**5**answers

840 views

### 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 ...

**2**

votes

**1**answer

198 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 $\mathbb{Z}^\ast$ ...

**3**

votes

**1**answer

213 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 ...

**2**

votes

**1**answer

339 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 $\prod_p \mathbb{F}_p/\mathcal{U}$?
...

**1**

vote

**3**answers

470 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?