**9**

votes

**1**answer

247 views

### Countable group with uncountable number of subgroups $< 2^{\aleph_0}$

Is it consistent that there is a countable group $G$ such that the cardinality of the set of subgroups of $G$ is uncountable, but strictly less than $2^{\aleph_0}$?

**10**

votes

**0**answers

182 views

### Generic $\mathbf{\Sigma}_3^1$-absoluteness for class forcings

In the paper "Generic Absoluteness" by Bagaria and Friedman (http://www.logic.univie.ac.at/~sdf/papers/bagfried.pdf) it is shown that in ZFC generic $\mathbf{\Sigma_3^1}$-absoluteness is false for ...

**7**

votes

**2**answers

298 views

### Topological tameness beyond the Gandy-Harrington topology

The Gandy-Harrington topology on $\omega^\omega$ is the topology generated by all lightface $\Sigma^1_1$ sets; that is, all sets which are continuous-in-the-usual-sense images of $\omega^\omega$.
...

**-1**

votes

**1**answer

110 views

### Non-standard naturals and goodstein sequences [closed]

By the Kirby–Paris theorem, Goodstein's theorem is independent of Peano arithmetic (PA). Therefore there are non-standard models in which every Goodstein sequence terminates. However, Tennenbaum's ...

**9**

votes

**1**answer

384 views

### Just a little absoluteness might be cheaper?

Absoluteness is a wonderful thing, but expensive consistency-strength wise. My question is, when can we get large amounts of absoluteness in specific situations for much cheaper?
Specifically, fix a ...

**8**

votes

**1**answer

192 views

### Is the Jordan decomposition of a self-adjoint functional constructive?

Let $A$ be an abstract C*-algebra, and let $\varphi\colon A \rightarrow \mathbb C$ be a bounded linear function. Assuming the axiom of choice, there exist unique positive bounded linear functions ...

**9**

votes

**0**answers

183 views

### The Chang model after collapsing an inaccessible limit of Woodins

If $\kappa$ is an inaccessible cardinal and $G \subset \operatorname{Col}(\omega,\mathord{<}\kappa)$ is a $V$-generic filter, then in $V[G]$ the Chang model $L(\text{Ord}^\omega)$ satisfies "every ...

**19**

votes

**3**answers

2k views

### Who needs Replacement anyway?

The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in ...

**4**

votes

**0**answers

184 views

### Primitive Closure Arithmetic

I am researching a system that was designed by myself and what I call PCA (Primitive Closure Arithmetic), because it looks like PRA.
The differences are:
- PRA uses recursive definition with a ...

**0**

votes

**1**answer

164 views

### Definition of “Expected/Unexpected Event”

Background of my question is Martin Gardner's "unexpected hanging" paradoxon, which has once again be the subject of an article in a popular-scientific magazin (this time because this year it has been ...

**11**

votes

**0**answers

255 views

### Adding a saturated ideal

Is it consistent that there is no $\omega_2$-saturated ideal on $\omega_1$, but one is introduced by an $\omega_2$-closed forcing?
Some motivation:
If $\delta$ is a Woodin cardinal, then it remains ...

**10**

votes

**0**answers

234 views

### c.c.c forcing notions and adding minimal generic reals

Is the following statement consistent:
``There is no non-trivial c.c.c forcing notion adding a minimal generic real''?
The question is related to Prikry's question: Is it consistent that any ...

**7**

votes

**1**answer

248 views

### For a ring $k$ and a set $X$, what are the $k$-algebra homomorphisms $k^X \to k$?

Let $k$ be a commutative ring. Feel free to assume it's a field.
Let $X$ be a set. This question is only interesting when $X$ is infinite.
Write $k^X$ for the $k$-algebra of functions $X \to k$, ...

**11**

votes

**3**answers

515 views

### The size of Lindelof space

Question. Suppose that $X$ is a Lindelof space such
that every point of $X$ is a $G_{\delta}$-point. Then is it true that $|X| ≤ 2^{\omega}$?

**9**

votes

**1**answer

389 views

### Is $\clubsuit_{\omega_1}$ enough to get Suslin tree?

This is problem 15.3 in Arnie Miller's problem list:
(Juhasz) Suppose there exists $\langle A_{\alpha} : \alpha \in L \rangle$, where $L$ is the set of limit ordinals below $\omega_1$ and for each ...

**2**

votes

**0**answers

86 views

### BL Algebras that allow for Compactness to hold

Say we have a model $M$ of a theory $T$ of some core fuzzy logic.
When dealing with compactness, we run in to a situation where the new model being built (by the use of compactness over $M$), will ...

**4**

votes

**1**answer

171 views

### Cohen algebra and $\mathcal P(\omega)/ \mathrm{fin}$

Let $C$ be the Cohen algebra, the boolean completion of the partial order of finite partial functions from $\omega$ to 2, ordered by reverse inclusion. Does there exist an ideal $I$ on $C$ such that ...

**11**

votes

**1**answer

201 views

### Does Peano's existence theorem admits a constructive proof?

$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$
The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...

**8**

votes

**2**answers

595 views

### What does “simplification of proofs as evaluation of programs” mean?

I am currently going through Philip Walder's "Proposition as Types" and a passage of the introduction has struck me:
for each way to simplify a proof
there is a corresponding way to evaluate a ...

**2**

votes

**0**answers

202 views

### Is there a $\Sigma^0_3$-complete ideal on $\omega$?

In Kechris's book "Classical Descriputive set theory" Chapter 23 (Exercise 23.4), it claim that there is a $\Sigma^0_\xi$-complete ideal on $\omega$ for each $\xi\geq 3$.
There is a candidate ...

**5**

votes

**1**answer

308 views

### Details for Woodin's forcing argument for a saturated ideal from the Levy collapse

Theorem 2.65 in Woodin's book shows that a saturated ideal on $\omega_1$ exists after Levy-collapsing a Woodin cardinal $\delta$ to $\omega_2$. I am confused about the part of the argument where he ...

**9**

votes

**1**answer

288 views

### Can there be a tree of height $\omega_2$ having all levels countable, with no cofinal branch?

For many years I had the idea that if a well-founded tree is both very tall and very narrow, then it must have a cofinal branch. For example, it is a fun exercise to show that any $\omega_1$-tree ...

**11**

votes

**0**answers

300 views

### On sentences true in all finite groups (revisited)

Let $w$ be a group word with variables $\bar x, \bar y$, where
$\bar x=(x_1,\dots ,x_m)$ and
$\bar y=(y_1,\dots ,y_n).$
I am interested in the following questions.
(1) Is the sentence $(\forall\bar ...

**7**

votes

**1**answer

193 views

### Introducing meets while preserving directed closure

A poset $\mathbb{P}$ is called well-met iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound.
Question: Suppose $\mathbb{P}$ is a separative partial order which is ...

**5**

votes

**0**answers

95 views

### O-Minimal sentences in $L_{\omega_1,\omega}$?

Is there any meaningful sense in which we can talk about o-minimal sentences of $L_{\omega_1,\omega}$? I can give a first attempt, easily; given a countable fragment $F$ and a sentence $\Phi$ in that ...

**22**

votes

**1**answer

938 views

### How hard is it to destroy a diamond? (with a real)

If we start with $V\models\lozenge$, it is not hard to force the failure of diamond. You can blow up the continuum, or destroy all the Suslin trees. You can blow up the continuum of $\aleph_1$, and ...

**7**

votes

**3**answers

276 views

### Do non-normal states exist in the Solovay model?

Let H be an infinite dimensional Hilbert space. Then there exist non-normal states on B(H) in ZFC (i.e. states that are not represented by a density operator).
Is this also true in the Solovay model ...

**1**

vote

**2**answers

159 views

### Are monoids with zero and partial homomorphisms related?

Context: Let $\Sigma=\{U,C,A,G\}$ and $L\subset\Sigma^*$, i.e. $L$ is a language over the alphabet $\Sigma$. Let $\Sigma'=\{0,1\}$ and define a homomorphism $f:\Sigma^*\to\Sigma'^*$ by extending $U ...

**38**

votes

**4**answers

1k views

### On sentences true in all finite groups

Let $w$ be a group word with two variables $x$ and $y$.
Is the sentence $(\forall x)(\exists y)w=1$
true in every group if it is true
in every finite group?
The same question about the sentence ...

**4**

votes

**1**answer

172 views

### Induction and nonstandard halting times of standard machines

For a nonstandard model of enough arithmetic - say, $\mathcal{N}\models I\Sigma_1$ - we can define the set of halting times of standard machines relative to $\mathcal{N}$: ...

**5**

votes

**2**answers

135 views

### ${\frak b}$ and ${\frak d}$ defined with $\leq$ instead of $\leq^*$

Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we define
$f\leq g$ if $f(n)\leq g(n)$ for all $n\in\omega$;
$f\leq^* g$ if there is $N\in\omega$ ...

**7**

votes

**0**answers

343 views

### Godel's second incompleteness theorem for non-r.e. theories

R. Jeroslow in this paper proves that a non-recursively enumerable theory whose set of theorems is $\Delta_2$-definable may prove consistency of itself
but it can not prove 2-consistency of itself.
...

**5**

votes

**1**answer

351 views

### Elementary equivalence of the direct product and direct sum of groups

It is well-known that the direct product of any family of abelian groups
is an elementary extension of the direct sum of the family
(see e.g. Lemma A.1.6 in the book `Model Theory' by W. Hodges,
...

**5**

votes

**1**answer

117 views

### Minimal degrees of structures

For this question, a structure means a first-order structure in a computable language with domain $\omega$; a copy of a structure $\mathcal{A}$ is a structure $\mathcal{B}\cong\mathcal{A}$.
Given a ...

**6**

votes

**1**answer

288 views

### If two structures are elementarily equivalent, is there a zigzag of elementary embeddings between them?

Fix a first-order signature $\Sigma$. There is an equivalence relation $\sim$ on the class $\Sigma\mathrm{-Str}$ of all $\Sigma$-structures given by $M \sim N$ iff $M$ and $N$ are elementarily ...

**2**

votes

**1**answer

167 views

### Explanation of the definition of Saturated Sets in Lambda Calculus

I have a question on the definition of Saturated Sets, as particular subset of the set of strongly normalizing terms in lambda calculus.
Here is the definition: a set $S$ of strongly normalizing ...

**29**

votes

**0**answers

758 views

### Bidi: A new cardinal characteristic of the continuum?

This question assumes familiarity with combinatorial cardinal
characteristics of the continuum.
Identify an infinite set $a\subseteq\mathbb{N}$ with its increasing
enumeration. Thus, for each natural ...

**1**

vote

**1**answer

302 views

### Is second-order ZFC categorical with regard to its proper class models

Second-order ZFC offers partial categoricity in the sense that, given any two models, one of them must be isomorphic to an initial segment of the other [1]. However, this leaves questions regarding ...

**2**

votes

**1**answer

192 views

### Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2

While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see [1]), these axioms tend to be somewhat arbitrary (e.g. adding an ...

**1**

vote

**1**answer

105 views

### PA proves that functions are total

Is there a total recursive function $f:N \to N$ such that for no $\Sigma_1$ formula $\phi(x,y)$ which defines it (i.e., defines its graph), is it true that PA proves that "$\phi$ defines a total ...

**4**

votes

**0**answers

117 views

### Is the lowenheim-skolem number of nth order logic larger than the corresponding number for 2nd order logic

According to this paper, by Vaananen, the $LS$ number for $2^{nd}$ order logic is given by "the supremum of $Π_{2}$-definable ordinals", where "The Lowenheim-Skolem number $LS(L)$ of $L$ is the ...

**4**

votes

**1**answer

226 views

### What consistency results follow the assumption: $\forall\alpha(\aleph_{\alpha+1}\nleq2^{\aleph_\alpha})$?

In a recent question on Math.SE it was asked whether or not For every infinite cardinal $\mathfrak m$ there is no $\aleph$ number, $\kappa$, such that $\mathfrak m<\kappa<2^{\mathfrak m}$.
By ...

**17**

votes

**4**answers

687 views

### Is there a Leibnizian model with no definable elements, in a finite language?

A first-order structure $M$ is Leibnizian, if any two distinct
elements $a,b\in M$ satisfy different $1$-types; that is, if there
is some formula $\varphi$ such that $M\models\varphi(a)$ and
...

**15**

votes

**1**answer

441 views

### Whatever happened to $L(j)$?

So this question probably shows my inner model theoretic ignorance, but:
In "Two remarks on elementary embeddings of the universe" (http://projecteuclid.org/download/pdf_1/euclid.pjm/1102969567), ...

**1**

vote

**0**answers

125 views

### What can be said about a Boolean-valued structure from what the Boolean-valued forcing extension thinks about it?

Suppose that $\phi$ is a formula in the language of set theory such that
there are some $n_{1},...,n_{k}$ such that if $V\models\phi(x)$, then $x=(X,R_{1},...,R_{k})$ and ...

**11**

votes

**0**answers

345 views

### Decidability of $x^3+y^3+z^3 = c$

I wondering if it is known whether the following problem is algorithmically decidable or undecidable by Turing machines: given an integer c, determine if there are integers $(x,y,z)$ such that ...

**6**

votes

**0**answers

257 views

### Reference for “if the set $A$ is Suslin, then every $\Sigma^1_1(A)$ set is Suslin”

Does anyone know of a reference for one or both of the following facts (in $\mathsf{ZF}$)?
If the set of reals $A$ is Suslin, then every $\Sigma^1_1(A)$ set of reals is Suslin.
If $T$ is a tree on ...

**13**

votes

**2**answers

905 views

### When does Vopěnka's principle hold?

Vopěnka's principle (VP) is the statement that, given any proper class $\{\mathcal{A}_\eta: \eta\in ON\}$ of first-order structures in the same language, there are some $\alpha\not=\beta$ with ...

**7**

votes

**1**answer

369 views

### Explicit counter example to Vopěnka's principle in the constructible universe?

Vopěnka's principle is a large cardinal axiom which has many equivalent formulations. One of them, which I find especially appealing, is the following: if the universe is satisfies Vopěnka's principle ...

**5**

votes

**1**answer

100 views

### When is a formula preserved under taking factors in a reduced product or the stalk in a Boolean product?

I want to know if there is a nice characterization of when a formula is preserved under taking reduced factors.
We say that a formula $\phi$ is closed under taking reduced factors if whenever $I$ is ...