**3**

votes

**0**answers

144 views

### 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 ...

**6**

votes

**1**answer

158 views

### 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.)
...

**2**

votes

**1**answer

119 views

### 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 $\Pi_1^0$ ...

**2**

votes

**0**answers

119 views

### 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 ...

**4**

votes

**1**answer

197 views

### Spectrum Problem for Higher-Order Logic

Definitions. Given a sentence $\varphi$ of $n$th-order logic, we define the spectrum of $\varphi$ to be the set of cardinalities of finite structures that satisfy $\varphi$. A set $X\subseteq\mathbb ...

**2**

votes

**1**answer

90 views

### Existentially closed partial orders

Existentially closed linear orders are dense linear orders without endpoints, which are finitely axiomatizable, and occur as order-types of natural mathematical structures such as the rationals or ...

**7**

votes

**1**answer

136 views

### Is there a Fraisse limit whose automorphism group contains dense but not generic automorphisms?

It is well known that $\mathsf{Aut}(\mathbb{Q},<)$ has generic automorphisms (i.e., a comeagre conjugacy class under the diagonal action) but does not admit ample generics. The automorphism group $\...

**10**

votes

**0**answers

148 views

### 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 $\aleph_1$...

**6**

votes

**3**answers

461 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 ...

**10**

votes

**1**answer

312 views

### 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 $T$. ...

**2**

votes

**0**answers

80 views

### 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 ...

**3**

votes

**0**answers

104 views

### Limits of definable maps

For sequences of semialgebraic maps there is the following result:
Let $(f_{n}: ]0,1[^d \to ]0,1[)_{n \in \mathbb{N}}$ be a sequence of continuous semialgebraic maps of bounded degree such that $(...

**2**

votes

**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, ...

**2**

votes

**0**answers

41 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 ...

**4**

votes

**1**answer

149 views

### Expressive power of $\omega$-order logic

According to the article Second-order and Higher-order Logic
from the Stanford Encyclopedia of Philosophy,
there is no need to stop at second-order logic; one can keep going. [...] we can allow ...

**10**

votes

**1**answer

325 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 x\;\...

**26**

votes

**4**answers

2k views

### Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic

Mathematician Edward Nelson is known for his extreme views on the foundations of mathematics, variously described as "ultrafintism" or "strict finitism" (Nelson's preferred term), which came into the ...

**1**

vote

**1**answer

143 views

### $\mathcal A\equiv\mathcal B\implies \mathcal A\cong\mathcal B$ for finite $\mathcal L$-structures where $\mathcal L$ is an infinite signature

Let $\mathcal L$ be an infinite signature and $\mathcal A$, $\mathcal B$ two finite $\mathcal L$-structures such that
for each first-order $\mathcal L$-sentence $\varphi$, $$\mathcal A\models\varphi\...

**2**

votes

**2**answers

165 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 ...

**17**

votes

**1**answer

403 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

368 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

164 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, \...

**1**

vote

**2**answers

282 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

583 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 ...

**25**

votes

**3**answers

738 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

247 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 ...

**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

290 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 ...

**23**

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

335 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

274 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

191 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

476 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 about,...

**6**

votes

**2**answers

224 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

299 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 $P:...

**9**

votes

**3**answers

558 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

135 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

169 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 $B$...

**19**

votes

**1**answer

696 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 ...

**24**

votes

**5**answers

2k 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

300 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

173 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

667 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

165 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

228 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

246 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

361 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

189 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 ...