**2**

votes

**2**answers

91 views

### Definable curves in definable sets

Suppose that I have an unbounded subset $X \subset \mathbb{R}^n$, definable in the $o$-minimal structure $\mathbb{R}_{an, exp}$. Is it possible to find an unbounded, analytic and definable curve (i.e. ...

**11**

votes

**6**answers

3k views

### Is there a relationship between model theory and category theory?

According to Chang and Keisler's "Model Theory", Model Theory = Universal Algebra + Logic. Model theory generalized Universal Algebra in the sense that we allow relation while in Universal Algebra we ...

**16**

votes

**1**answer

393 views

### 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 ...

**14**

votes

**0**answers

323 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 ...

**4**

votes

**2**answers

153 views

### Can one satisfaction class code another?

Let $M$ be a model of ${\sf ZFC}$. A satisfaction class $S$ for $M$ is subset of $M$'s ordered pairs which satisfies in $M$ the standard Tarskian compositional axioms. E.g.:
$M\vDash \forall \phi, ...

**0**

votes

**2**answers

276 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 ...

**12**

votes

**1**answer

557 views

### 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 ...

**10**

votes

**1**answer

288 views

### Uniform elimination of imaginaries

Does the following principle follow from uniform elimination of imaginaries?
For every formula $\varphi(x;y)$ there is a formula $\vartheta(x;z)$ such that
$$\forall y\;\exists^{=1}z\;\forall ...

**1**

vote

**0**answers

29 views

### Quantifier elimination of pp-subgroups of modules

This is a model-theoretic questions. Let $M$ be a $R$-module. Our language will be the standard language of modules, i.e. the language of abelian groups together with an unary function symbol for ...

**26**

votes

**3**answers

699 views

### Does the category of (algebraically closed) fields of characteristic $p$ change when $p$ changes?

Let $\mathrm{ACF}_p$ denote the category of algebraically closed fields of characteristic $p$, with all homomorphisms as morphisms. The question is: when is there an equivalence of categories between ...

**4**

votes

**2**answers

219 views

### Least inner model of ZF without power set axiom

I'm interested in the existence and properties of an analogue version of $L$ for models of ZF$^-$ (ZF without the power set axiom), which for simplicity I'll call $L^-$. By "analogue" I mean the least ...

**1**

vote

**6**answers

2k views

### Looking for a source for Intended Interpretation

Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, ...

**17**

votes

**2**answers

2k views

### 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 ...

**10**

votes

**0**answers

273 views

### A question concerning model theory of groups

Several days ago, Professor Martin Bridson gave a very nice talk in my department. Some questions concerning his talk came into my brain Since I am neither a model theorist nor a algebraist, I am not ...

**22**

votes

**9**answers

3k views

### What is… A Grossone?

Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The mathematical value of this ...

**1**

vote

**1**answer

327 views

### 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 ...

**6**

votes

**1**answer

269 views

### Are there first order theories of interest to an algebraist or at least a model theorist of large cardinal consistency strength?

I am wondering if there are some first order theories of algebraic structures or structures of interest to model theorists of large cardinal consistency strength or at least unexpectedly high ...

**8**

votes

**0**answers

187 views

### ω-categorical, ω-stable structure with trivial geometry not definable in the pure set

Briefly, my question is the following.
does every countable ω-categorical, ω-stable structure with
disintegrated strongly minimal sets interpret in the countable pure set?
By countable pure set I ...

**6**

votes

**1**answer

470 views

### Godel's proof of Completeness

Where could I find a detailed exposition in English of Godel's proof (not Henkin's) of Completeness Theorem for first order logic? The wikipedia article omits certain details that I am not clear ...

**6**

votes

**2**answers

208 views

### Examples of NIP fields of characteristic $p$

Definition. According to Shelah, a field $K$ does not have the independence property (i.e. is NIP) if for every first order formula $\varphi(x, \bar y)$ in the language of fields $(+,\times,0,1)$, the ...

**9**

votes

**0**answers

263 views

### Elementary proof of Chevalley's Theorem on constructible sets

I am looking for a proof of the easiest affine version of Chevalley's Theorem on constructible sets :
Theorem (Chevalley). The image of a constructible subset of $\mathbf C^n$ by a polynomial map ...

**9**

votes

**3**answers

551 views

### 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.
...

**5**

votes

**0**answers

127 views

### 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 ...

**5**

votes

**1**answer

165 views

### 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 ...

**19**

votes

**1**answer

634 views

### 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 ...

**23**

votes

**5**answers

1k views

### 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 ...

**2**

votes

**1**answer

1k views

### 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 ...

**5**

votes

**1**answer

288 views

### Constructive compactness for countable models?

The compactness theorem for countable (Tarski?) models is equivalent to the weak König's lemma by a result of H. Friedman and others as noted here, in the context of classical logic. The weak König's ...

**1**

vote

**0**answers

170 views

### Reference request: Models of isomorphic languages result into isomorphic categories

This is basically a reference request by someone who has not been educated as a logician and would like to be rigorous about certain preliminary aspects of model theory.
Fix an uncountable universe ...

**18**

votes

**2**answers

660 views

### 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 ...

**4**

votes

**0**answers

159 views

### Hyperimaginaries and continuous logic

Classical (i.e., discrete) logic is well positioned to study imaginaries in part because the $T^{eq}$ construction allows us to treat imaginary sorts as we would treat any other sort. With ...

**4**

votes

**1**answer

223 views

### When does an infinite model have a proper class-sized elementary extension?

Suppose that a set of sentences of a 1st order language has an infinite model $M$.
Under what conditions is there is a proper class-sized elementary extension of $M$?
How does the answer change if ...

**3**

votes

**2**answers

242 views

### References for the Keisler Order

Are there any good/modern references on the Keisler order. I have been reading Keisler's original paper, "Ultraproducts which are not Saturated", which introduces the order. However it is somewhat ...

**9**

votes

**0**answers

351 views

### Riemann hypothesis in Zilber's field

Question. What is known about the situation (truth or falsity) of Riemann hypothesis in the Zilber's field?

**3**

votes

**1**answer

184 views

### Compactness in Bishop's constructive mathematics

In Bishop's constructive mathematics, is there any literature on a possible version of the weak König's lemma, or of the compactness theorem for countable models? There is some related information ...

**11**

votes

**1**answer

292 views

### Intutionistic Robinson Arithmetic

By Friedman translation $HA$ and $PA$ prove the same $\Pi_2$ formulas.
Is it true for Intutionistic Robinson arithmetic(Robinson axioms with intutionistic logic) and classic Robinson arithmetic?
...

**9**

votes

**6**answers

669 views

### Applications of forcing in model theory

What are the major applications of (set theoretic) forcing in model theory?

**-2**

votes

**1**answer

249 views

### A set theory and a model without the empty set [closed]

Im just asking out of curiosity.
Is there a set theory $\mathcal T$ and a model $\langle M,E\rangle \vDash \mathcal T$ such that no set in M is empty:
$$\left(\forall m\in M \right)\left(\exists n\in ...

**24**

votes

**1**answer

716 views

### Do the analogies between metamathematics of set theory and arithmetic have some deeper meaning?

By "formal analogies" between the metamathematics of $\mathsf{ZFC}$/set theory and $\mathsf{PA}$(=Peano Arithmetic)/first order arithmetic, I mean facts such as the following:
We are considering a ...

**2**

votes

**0**answers

64 views

### models of $I\exists^+_1$

$\phi$ is a $\exists^+_1$ formula iff it is in language of arithmetic and does not have $\forall$,$\neg$ and $\rightarrow$, therefore $I\exists^+_1$ is theory of $Q$+induction axioms for $\exists^+_1$ ...

**4**

votes

**0**answers

172 views

### Relation between fundamental theorem of Ehrenfeucht-Fraisse games and notions of bisimulation and simulation

Vijay D alluded to the relation between the fundamental theorem of Ehrenfeucht-Fraisse games and notions of bisimulation and simulation in response to the whats-a-magical-theorem-in-logic question.
...

**3**

votes

**1**answer

110 views

### Scott Rank of Models of Infinitary Sentences

Let $\mathscr{L}$ be a recursive language. Let $\varphi$ be a $\mathscr{L}_{\omega_1 \omega}$-sentence and $\varphi \in L_{\omega_1^\emptyset}$. (Let $\varphi$ be a computably infinitary formula.) Let ...

**2**

votes

**1**answer

114 views

### How to show certain theories are not existentially closed

To show that $ZFC$ is not existentially closed, we can use the following forcing argument: Let the ground model can model $V=L$ and the forcing extension model $2^{{\aleph}_{0}}=\aleph_{2}$. (Maybe ...

**5**

votes

**0**answers

83 views

### Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$?

Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$?
In End extensions of models of linearly bounded arithmetic paper the author said this problem is open. I ...

**3**

votes

**0**answers

91 views

### Does Łoś's theorem imply choice given a free ultrafilter?

In the paper "Łoś's theorem and the boolean prime ideal theorem imply axiom of choice" Howard has shown that Łoś's theorem and the boolean prime ideal theorem imply axiom of choice. At the end of the ...

**7**

votes

**2**answers

205 views

### How to extend Morley's omitting type theorem to uncountable languages?

In his 1965 paper Omitting Classes of Elements (found in The Theory of Models: Proceedings of the 1963 International Symposium at Berkeley, published by North-Holland Publ. Co., Amsterdam (1965)), ...

**3**

votes

**1**answer

150 views

### When can we have “each subtheory is satisfiable iff it is recursively axiomatizable”?

Weak Version:
Is there a 1st order language $L$ (with only countably-many formulas) such that for each recursive coding $C$ of the formulas of $L$, there is a theory $T$ of $L$ where
$T$ is not ...

**13**

votes

**1**answer

333 views

### Is this theory decidable?

It is well-known that both Presburger arithmetic (by contrast with Peano arithmetic) and Tarski geometry are decidable. I was in the shower this morning and wondered whether there exists an elegant ...

**5**

votes

**1**answer

307 views

### A proper class of formulas with every set-sized (but no proper-class-sized) subcollection satisfiable

What feature(s) must a (non 1st-order) language with proper-class-many formulas have in order to guarantee that:
There is a proper class P of formulas such that both
(a) every set-sized ...

**4**

votes

**3**answers

217 views

### End Extension models of $I\Delta_0$

Recently I'm thinking about question below, but I can not prove or disprove it.
Is it true that for every model $M\models I\Delta_0$ there exists a
model $M'\models PA$ such that $M'$ is end ...