**11**

votes

**1**answer

430 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

331 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

189 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?

**4**

votes

**2**answers

431 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

265 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

204 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

152 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

497 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

622 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 ...

**6**

votes

**0**answers

311 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

368 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

321 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

554 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

517 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

994 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

797 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

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

**3**

votes

**1**answer

212 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

336 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

430 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?

**15**

votes

**0**answers

786 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 ...

**2**

votes

**0**answers

431 views

### 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 ...

**4**

votes

**4**answers

919 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 ...

**2**

votes

**0**answers

287 views

### 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 ...

**4**

votes

**4**answers

897 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 ...

**4**

votes

**1**answer

711 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 ...

**5**

votes

**1**answer

318 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 ...

**1**

vote

**1**answer

321 views

### Real closed field+ model thoery

Is it true that every real closed field can be elementarily embedded in some other real closed filed with the same Archimedean classes (I mean in a proper extension)?
Can for example real numbers be ...

**5**

votes

**4**answers

675 views

### Example of two structures

This is probably a very trivial question, still I don't seem to find an answer.
I'd like to see an example (in some language) of two countable structures $\mathcal{M}_1 $ and $ \mathcal{M}_2 $ with ...

**5**

votes

**1**answer

346 views

### Determining the exceptional set in the theorem of Ax & Kochen

Ax & Kochen [1] proved that for every $d\in\mathbb{N}$ there exists a finite set $A(d)$ such that for every prime $p\not\in A(d),$ every homogeneous polynomial of degree $d$ over $\mathbb{Q}_p$ in ...

**7**

votes

**3**answers

905 views

### Is the set of undecidable problems decidable?

I would like to know if the set of undecidable problems (within ZFC or other standard system of axioms) is decidable (in the same sense of decidable). Thanks in advance, and I apologize if the ...

**0**

votes

**1**answer

152 views

### Interdefinability of two expansions of the Real Field

I was asked the following question two days ago, but I couldn't completely resolve it.
Here is the claim:
$\mathcal R = (\mathbb R,+,\cdot)$ is the real field.
Let $I$ be an open interval (perhaps ...

**4**

votes

**2**answers

319 views

### 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 ...

**3**

votes

**0**answers

157 views

### Intersecting the algebraic closure of independent elements

$G$ is a group with a simple first order theory $T$ as defined by Shelah, hence equiped with a "nice" notion of independence. $G$ also has generic elements. I write $acl^n(A)$ for the set of elements ...

**8**

votes

**2**answers

528 views

### are there standard examples of stable theories that are undecidable?

What is known about decidability of various first order theories studied in stable model theory, geometric model theory, o-minimality? For example, is there a natural example of an undecidable ...

**12**

votes

**2**answers

686 views

### Connes' embedding conjecture for uncountable groups

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 ...

**10**

votes

**2**answers

702 views

### A Model-Theoretic Helly's Theorem

There is a combinatorial question posed to me (or rather, posed near me) by my adviser. I am having quite a lot of difficulty proving it. It goes:
For any NIP theory $T$ (complete with infinite ...

**3**

votes

**0**answers

365 views

### Showing that every satisfiable sentence with at most two variables has a finite model

I have tried to prove, in first order logic, that every satisfiable sentence (without function symbols) with at most two variables has a finite model. My attempts were unsuccessful.
This is an ...

**3**

votes

**1**answer

400 views

### Examples of Graphs with Trivial Definable Closure

If I'm giving a lecture on trivial definable closure (as a property of graphs), what is a good example of a graph that I can easily draw to depict the concept of trivial dcl?

**2**

votes

**2**answers

433 views

### Definable subsets of the integers as an abelian subgroup?

Consider the integers as a first-order structure in the language {0,+,-} of abelian groups. I suspect that the collection of definable subsets (without parameters) of this structure is an algebra ...

**2**

votes

**1**answer

267 views

### Extending elementary maps in stable theories.

Take some stable theory $T$ with elimination of imaginaries, all sets appearing are small subsets of the monster model of $T$, elementary maps are restrictions of automorphism of the monster model of ...

**5**

votes

**1**answer

1k views

### Definable measure preserving isomorphisms of $p$-adic semialgebraic sets

Hi,
Consider a $p$-adic field $K$ (finite extension $\DeclareMathOperator{\bQ}{\mathbb{Q}}$of $\bQ_p$) in Macintyre language $\DeclareMathOperator{\cL}{\mathcal{L}}$ $\cL_{\rm Mac}$. Let $Z$ be a ...

**7**

votes

**2**answers

363 views

### New results on Chow's notion of closed-form numbers?

In an interesting article (available here), Timothy Chow proposes that a closed-form number be defined as an element of the smallest subfield of $\mathbb{C}$ that is closed under $\exp$ and a chosen ...

**4**

votes

**6**answers

2k views

### A book about model theory

I am looking for a good book about model theory. As this is obviously too vague, let me
explain what I am looking for and why.
First I am interested about the basics and foundations of model theory. ...

**2**

votes

**2**answers

289 views

### “Duals” of Lindenbaum algebras

From Wikipedia I learn:
The Lindenbaum algebra A of a theory T consists of the equivalence
classes of sentences of T. The operations in A are inherited from those in T.
If there are ...

**4**

votes

**2**answers

788 views

### Are all countable, nonstandard models of arithmetic given by ultrapowers?

Countable models of PA fall into two categories: the standard one $(\omega, S)$ and the nonstandard ones (all the rest). The only way I've seen to construct a nonstandard model is through taking an ...

**23**

votes

**15**answers

4k views

### 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 ...

**4**

votes

**2**answers

336 views

### Stable theory: question about definability of independece

I would like to know why the relation: R(x,y) iff x independent from y (i.e: tp(x/y) doesn't fork over the empty set) is type definable in stable theory?
Thanks to the helper.

**2**

votes

**1**answer

394 views

### Ergodic Invariant Measures and the Rado graph

Given two or more invariant measures on a structure, there are various ways to combine them to form another invariant measure on the structure. For example, given two invariant measures on a ...