**6**

votes

**2**answers

232 views

### Is every non-recursive set in $\Sigma_1$ complete in $\Sigma_1$ (relatively to many-to-one reductions)?

Most well known sets in $\Sigma_1 \setminus\Delta_0$, such as the
Halting problem, are complete in $\Sigma_1$, relatively to the
many-to-one reduction. In fact I don't know any example of a (non ...

**1**

vote

**0**answers

56 views

### A question on completeness for quantified temporal logics

Quantified temporal logics have the Barcan formulas and its converses for both G (it will always be the case that) and H (it has always been the case that), so that both $\forall x G \alpha ...

**5**

votes

**1**answer

247 views

### Woodin Cardinals and Inner Models

I have a few questions I have been thinking about that I could definitely use some insights on:
Question 1. Since a Woodin cardinal is a "local" notion, defined with respect to some rank-initial ...

**1**

vote

**1**answer

102 views

### Preserving Predimension Functions under Functional Convergences

Definition 1. If $\mathcal{L}$ is a countable relational language, a predimension class $C$ is a class of $\mathcal{L}$-structures with the following properties:
C1: ...

**2**

votes

**0**answers

115 views

### Preimages of accessible full subcategories

My question is ultimately about the model theory of $L_{\infty, \omega}$, but it is more convenient to phrase it in terms of category theory. Suppose I have finitely accessible categories ...

**6**

votes

**1**answer

186 views

### Is a computer program for correspondence theory available?

In the 1990s I some times used a computer program with the Max Planck Institute which helped with calculating complicated correspondences for modal logical formulas. Is some program like that ...

**4**

votes

**1**answer

152 views

### Stable examples from Algebra such that the model theoretic algebraic closure of a substructre is no model

Let $T$ be a stable theory. Let $A$ be a subset or substructure of a model $M$ of $T$. Now in some theories the (model theoretic) algebraic closure of $A$ is already a (sub)model of $T$. For example, ...

**1**

vote

**0**answers

39 views

### Completeness results for quantified tense logics with BF?

Modal tense logics or temporal logics are important in that they correspond with partial orders and their extensions.
Are there completeness results for quantified temporal logics with the Barcan ...

**4**

votes

**0**answers

103 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.
...

**9**

votes

**3**answers

356 views

### Is a model of arithmetic contained in a model of arithmetic an initial segment?

It's easy enough to show that if $\mathbb{N}_1$ is a non-standard model of the Peano axioms, then there is a canonical embedding $\mathbb{N} \to \mathbb{N}_1$, and we have a theorem that if $x \in ...

**7**

votes

**3**answers

382 views

### Decidability of the winning-position problem in an infinity chess with a finite number of short-range pieces only

Definitions
Long-range pieces: queens, rooks, bishops.
Short-range pieces: pawns, knights, kings.
We can extend the definition of short-range pieces to include also fairy pieces like: Berolina ...

**16**

votes

**2**answers

473 views

### Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point.
Is the notion of FPP for topological spaces an absolute notion? More ...

**3**

votes

**0**answers

51 views

### A question on the incompleteness of quantified K.2 and S4.2 with the Barcan formula

I have been attempting to come to grips with Max Cresswell's account of this in Journal of Philosophical Logic 24 (4):379 - 403 (1995) where he presents proofs of the incompleteness of QK.2BF as well ...

**7**

votes

**0**answers

347 views

### Is there a theory of abuse of notation? [closed]

Is there any theory about the different ways notation can be abused and which abuses are ineliminable without complicating the notation in some essential way? We can define "abuse of notation" as any ...

**1**

vote

**2**answers

243 views

### Infinite play with tape, or covering the integers with prime arithmetic progressions

It is possible that a more technical version of this question has been
asked and answered in the literature. If so, then a reference is much
appreciated. I will phrase it in terms of colored tapes ...

**2**

votes

**1**answer

130 views

### A question on the modal logic S4.2

The modal logic S4.2 with the characteristic axioms
4: $\square \alpha \rightarrow \square \square \alpha$
and
.2: $\lozenge \square \alpha \rightarrow \square \lozenge \alpha$
and
T: ...

**10**

votes

**0**answers

160 views

### Maximality of linear orders in the Keisler order on theories

Recently Malliaris and Shelah (see their preprint http://math.uchicago.edu/~mem/Malliaris-Shelah-CST.pdf) have shown that theories with $SOP_2$ are maximal in the Keisler order. A preceding result of ...

**5**

votes

**1**answer

118 views

### Attribution of an equivalence of the existence of omega-models of RCA0

There are many well-known equivalences in reverse mathematics between statements of the form "Every set is contained a countable coded $\omega$-model of $T$" and $S$, where $S, T$ are subsystems of ...

**6**

votes

**1**answer

165 views

### Is (Z,+,0,1,P2,P3) decidable?

Is Presburger arithmetic, augmented with two unary predicates P2, P3, for powers of 2 and powers of 3 respectively, decidable?
I know that adding just one of P2, P3 to Presburger keeps it decidable, ...

**1**

vote

**1**answer

405 views

### Compactness and completeness in Gödel logic

The standard proof of the completeness theorem in first-order Gödel logic is based
on a first-order countable language. I want to know that is there any proof of the completeness theorem in ...

**16**

votes

**2**answers

483 views

### Internal logic of the topos of simplicial sets

I am looking for a closed statement (i.e. not depending on any parameter objects) which is true in the internal logic of the topos of simplicial sets, but is not an intuitionistic tautology. Ideally, ...

**8**

votes

**3**answers

625 views

### What set theoretical questions could never be answered by Turing machines of arbitrary cardinality?

Let us assume that there are Turing machines of arbitrary cardinality, by that I mean they can have input tapes of any arbitrarily high cardinality and compute for a number of steps also of ...

**12**

votes

**1**answer

200 views

### Nontrivial upper bounds on proof-theoretic ordinals of strong theories: do we have any?

Motivated by Consistency of Analysis (second order arithmetic) and Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?, I have the following question:
Are there any examples of strong ...

**10**

votes

**5**answers

2k views

### Interesting meta-meta-mathematical theorems?

The Goedel incompleteness theorems can be considered meta-mathematical theorems, as they are "written" in a meta-theory and "talk" about properties of a class of formal theories.
The following may be ...

**3**

votes

**0**answers

144 views

### Equational theories determined by “identities without variables”

How to characterize equational theories $T$ which have the following property: for any two terms $t(x_1,...,x_n)$ and $t'(x_1,...,x_n)$ in the signature of $T$, if for any closed terms (i. e. terms ...

**8**

votes

**3**answers

634 views

### Consistency of Analysis (second order arithmetic)

Is there a proof of the consistency of Analysis (second order arithmetic), which is similar to Gentzen's proof of the consistency of arithmetic?
Update:
Which (different) methods can be used to ...

**3**

votes

**1**answer

233 views

### Colorful model theory

There are a number of concepts in model theory - often situated around Hrushovski's amalgamation method (see for instance http://math.univ-lyon1.fr/~wagner/nijmegen.pdf) - which are colorfully named:
...

**4**

votes

**1**answer

221 views

### Perfect set property implies $\omega_1$ is a limit cardinal in $L$

Specker proved in 1957 that if in $V$ every set of real numbers has the perfect set property, than in $L$, $\omega_1^V$ is actually a limit cardinal.
The original proof is in German, and I've been ...

**2**

votes

**0**answers

106 views

### Does the Lévy collapse obey this nice characterization? [duplicate]

This question is related to an issue in my answer to Monroe Eskew's question on the failure of Cantor-Bernstein for the Lévy collapse.
Question. Is the Lévy collapse $\text{Coll}(\omega,\lt\kappa)$ ...

**6**

votes

**1**answer

173 views

### Failure of Cantor-Bernstein for the Levy Collapse

Related to this question, is it possible to give an example of the failure of Cantor-Bernstein for complete embeddings of forcing notions involving the Levy Collapse $Col(\omega,<\kappa)$? Suppose ...

**5**

votes

**1**answer

220 views

### A model of Krivine

In a paper by J.-L. Krivine, Modèles de ZF+AC dans lesquels tout ensemble de réels définissable en termes d'ordinaux est mesurable-Lebesgue [C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A549–A552, ...

**5**

votes

**1**answer

161 views

### What do algebraic theories with strictly terminal trivial models look like?

By algebraic theory I mean one in the sense of Lawvere, i.e. a collection of finitary operations, including projections, together with a multi-composition satisfying the obvious axioms. (I believe ...

**7**

votes

**1**answer

488 views

### Starting Hilbert's Program on the other end

The idea of Hilbert's program was to start with a simple finitary logic and proof the consistency of more complex systems in this system. Of course, this turned out to be problematic. Even when ...

**4**

votes

**1**answer

119 views

### Replace Morley sequence over some set by one over a finite set, s.t. they both satiesfy a certain formula

Let $T$ be a stable $L$-theory with elimination of imaginaries. We work in the monster model $\mathfrak C$ of $T$. Let $A$ be a small (infinite) set of the monster, $\phi(x,y)$ be a $L(A)$-formula and ...

**7**

votes

**2**answers

459 views

### What is the name of this type of groups?

Suppose $A$ is a finite set and $\Sigma=A\cup A^{-1}$. Let $L\subseteq \Sigma^{\ast}$ be a regular language on the alphabet $\Sigma$. Is there a common name for the group $G$ presented as:
$$G=\langle ...

**4**

votes

**1**answer

195 views

### Is Van der Waerden's function elementary

Van der Waerden's function was proved to have elementary upper bound on growth rate.
Is the Van der Waerden's function itself elementary in the sense of Kalmar?

**2**

votes

**1**answer

252 views

### Expressive power of first-order category theory

Given the signature $\lbrace \mathsf{dom}, \mathsf{cod}, \mathsf{id},\circ \rbrace$ and the axioms of category theory – which are expressible in the signature's first-order (FO) language – ...

**2**

votes

**1**answer

129 views

### How many elementary equivalent models are unifiable by ultrapower?

Definition. A class $\mathcal{C}$ of pairwise elementary equivalent $\mathcal{L}$-structures is unifiable by ultrapower if there is an index set $I$ and an ultrafilter $F$ on it such that $\forall ...

**9**

votes

**4**answers

1k views

### Are there non-diagonal proofs for Cantor's continuum and Godel's incompletness theorems?

There is a formal definition for the notion of a formal proof.
Question 1. Is there any formal definition for the notion of a diagonal formal proof?
Consider the following theorems both proved by ...

**1**

vote

**0**answers

59 views

### First-order Peano Axioms and order-completeness of $\mathbb{N}$ [closed]

Definition: An ordered set is order-complete if any nonempty subset with an upper bound, has a lowest upper bound or supremo.
Notation: We denote the system of first-order Peano Axioms (along with ...

**9**

votes

**1**answer

192 views

### Obtaining a lightface pointclass from a boldface one

Define a pointclass to be:
boldface inductive-like if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and ...

**10**

votes

**1**answer

272 views

### Reverse mathematics of meromorphic functions on Riemann surfaces

Various sources touch briefly on the reverse mathematics of measure theory and complex analysis. But I have found none on the uniformization theorem for Riemann surfaces or the existence of ...

**10**

votes

**3**answers

875 views

### Is it ever a good idea to use Keisler-Shelah to show elementary equivalence?

The most useful way I know to show that two structures are elementarily equivalent is Ehrenfeucht-Fraisse games. These are quite nice and intuitive, and even when I can't use them to solve my problem ...

**4**

votes

**1**answer

140 views

### Finite generation of vector identities

This question is partially motivated by Looking for a comprehensive referece for vector identities, although that question may not be appropriate for MO.
Consider the set $\mathcal{E}$ of all valid ...

**5**

votes

**1**answer

193 views

### A ZFC construction to get a proper extension which is a $\omega_1$-model

In $V$, let me call a set theory structure A is a $\omega_1$ model if the $\omega_1$ of $A$ is the same as the $\omega_1$ in $V$ (up to isomorphism). The question I would like to ask is the following: ...

**6**

votes

**1**answer

186 views

### Skolem Hulls in $H_{\omega_2}$

I put this on stack exchange over a week ago with no answer, so let's try here.
Consider a model of the form $\mathfrak{A} = (H_{\omega_2}, \in, \lhd, f_0, f_1, ...)$, some expansion of ...

**-2**

votes

**1**answer

185 views

### Natural constructions (not depending on parameters) [closed]

Consider graph clusterings as a prototypical example of (logical) constructions.
Let a clustering of a graph $(V,E)$ be any covering of $V$, i.e. a set $C$ with $\bigcup C = V$.
I am looking for a ...

**8**

votes

**2**answers

240 views

### Vaught's conjecture for partial orders

In
``Steel, John R. On Vaught's conjecture. Cabal Seminar 76–77, pp. 193–208''
the following is proved:
Theorem. Let $\phi\in L_{\omega_1,\omega}.$ If every model of $\phi$ is a tree, then $\phi$ has ...

**9**

votes

**3**answers

348 views

### Conjecture on NP-completeness of tesselation of Wang Tile up to finite size

Motivated by these following questions on tessellation:
coloring in lattice
Reference for Wang Tile
Computational approach deciding whether a set of Wang Tile could tile the space up to some size
...

**2**

votes

**3**answers

380 views

### Can a Decidable Theory Have Non-recursive Models?

Tennenbaums' theorem proves neither addition nor multiplication can be recursive in any countable non-standard model of arithmetic. Tennenbaum's proof applies to theories much weaker than PA.
...